Introduction

World needs the energy for endurance of the lives to address the global energy demands while minimizing the environmental impact, the integration of solar energy into power systems is a pivotal solution, Solar Energy systems. Solar energy’s vital role as a renewable source has led to extensive research in enhancing the efficiency of PV systems through improved materials designs and integration with energy management systems1. Advancements in Solar photo voltaic (PV) techniques have significantly transformed energy generation establishing solar power as an incredibly sustainable alternative to conventional energy resources and more viable for various applications2. The review of the global and solar energy status across 235 countries and territories highlights solar energy contributions to global electricity production are typically 3.6\(\%\) which is low. Academic research helps in the shaping of the solar analysis industry and the potential for growth in 30 countries accounting for 12.8\(\%\)of the survey nations. Remarking the challenges includes the intermittency of solar power generation due to complexity in the system design and control. Additionally, the economic factors such as initial investment costs3. The design of PV arrays, boost converters, and voltage sources is essential for efficient energy transfer and grid integration and by overcoming the challenges of variability in solar by weather change, and efficiency of power conversions4. Solar Energy systems, enable more resilient and sustainable power infrastructure, which has driven the development of advanced technologies, including the MPPT controllers, to optimize power output under varying conditions, and enhance the overall efficiency of a solar power system. Incremental Conductance (INC) and Perturb and Observe (P&O), are crucial for extracting the maximum possible power from the solar arrays. Integrating the INC method with the boost converter enhance the responsiveness ensuring optimal performance and maximizing energy harvest from the solar panels2. DC-DC converters are essential for converting direct current (DC) electrical power to different levels required for the applications. DC-DC converters operate on the principles involving inductors, capacitors, and switches to control the energy flow. These require effective control strategies for optimal performance such as voltage mode control and the current mode control. Advanced technologies like adaptive and predictive control further enhance the converter performance by adding real-time parameters based on the load conditions and the input voltage variations2. DC-DC boost converters are essential in the photovoltaic system, enhancing the voltages from the solar panels for grid integration. Advanced control techniques like feedback linearization and the disturbance observers (DO) improve their performance and simulation tools like Matlab or Simulink are used to model and analyze the experimental results to validate their effectiveness and give insights into the optimal design which focuses on the real-time implementation and adaptive control strategies to address the dynamic nature of the solar analysis generation5. The integration of renewable energy, particularly the solar PV system into microgrids has significantly gained attention due to the demand for sustainable energy solutions. Advanced control techniques such as model predictive control (MPC)6enhance the performance of converters like Voltage Multiplier Coupled Cascaded Boost Converters7considering the future load demands and energy pricing to optimize the power flow and minimizing the operational cost. AC-coupled system architecture simplifies the integration of various energy sources and the storage system but may not be as significant as a DC-coupled system, which involves fewer conversion stages but requires complex control strategies8.

Fig. 1
figure 1

Hybrid Microgrid Schematic Diagram.

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Advanced pulse with modulation (PWM) techniques have been developed to reduce the total harmonic distortions (THD) and switching losses. Despite the progress, challenges remain in integrating renewable energy into the microgrids the high installation and repair cost repost control strategies to manage fluctuating enhances and integrating various energy storage technologies and microgrid bus voltage stability requires advanced control techniques and robust converter designs to ensure the reliability and stable operation of a solar system and its integration into the grid8. Adaptive control strategies like active damping and sliding mode control (SMC)8in bidirectional DC-DC converters in solar PV systems1,2facilitate efficient energy management between the solar array, the battery storage, and loads6 which can improve the system’s stability and response to disturbances. Power electronic converters have improved energy conversion efficiency, with DC-DC Converters playing a crucial role in managing the energy within hybrid microgrids as shown in Figure 1. Different types of topologies include isolated, resonant, and hybrid converters. Isolated converters enhance safety and allow high voltage transformations, whereas resonant converters utilize high efficiency and reduce electromagnetic inference (EMI), and Hybrid converters combine different topologies to optimize performance. The two-stage hybrid isolated DC-DC boost converters can handle the wide input voltage and maintain high efficiency9. Conventional boost converter is widely recognized for its simplicity and effectiveness in low-power applications. Still, their limitations in high-power scenarios due to inductor saturation have led to the development of more advanced converters such as interleaved boost converters (IBC), which offer improved performance by reducing the voltage and the current ripples while increasing the current ratings8. High step-up gain DC-DC converter combines the switched capacitor and regenerative boost configuration to achieve high DC voltage gain without bulky transformers10. However, the limitations are simplifying complex converter design while ensuring high performance and maintaining high efficiency under varying conditions, and managing electromagnetic inference, which can be modified by soft-switching characteristics1. High-boosting DC-DC converters increase the output while maintaining efficiency by reducing the number of series connections of PV modules. On the maximum power point tracking algorithms optimize the energy extraction. Cascading DC-DC converters can be used to improve PV performance from the partial shielding effect11. The recent advancements in the converter topologies such as dual inductor and hybrid boosting converters which aims to tackle the limitations of the current converters while enhancing its performance in terms of efficiency and voltage gains12. Transformers-less inverters offer high efficiency, reduced weight, and lower cost making the perfect use for residential applications11. High-frequency magnetic link-based converters offer lower power losses and better insulation6. Isolated solar power generation systems are particularly beneficial for off-grid or remote applications, by incorporating the batteries as energy storage solutions. For longevity of solar power systems13. The rise of small-scale solar applications has led to innovation in energy storage solutions, such as lithium-ion batteries which store the excess energy for continuous supply during low irradiation periods1. DC-DC converter applications such as in EVs, manage the power distribution between the battery and the electric motor. These play a crucial role in consumer electronics and industrial automation for providing stable power to various components and systems. With the recent advancements in solar power generation systems which aims to reduce greenhouse emissions and enhance efficiency the adoption of the small capacity distributed path generation systems particularly in residential areas helps to accelerate due to reduce the cost of the solar panels. The power conversion interface converts the intermittent DC power to the AC power for the grid integration14. As discussed through the works15,16,17,18,19,20, the energy systems consisting the power conversion interface has been widely used in microgrids, which moves to proceed the current research.

Motivation

The PV system is booming because of the urgent need for more efficient sustainable and accessible energy solutions. Innovations like bifacial solar panels, which capture the sunlight on both sides to boost the energy output, and the perovskite solar cells known for their high efficiency and flexibility are at the lead of this transformation. Developing double-boosting techniques in PV systems lies in achieving higher efficiency, greater voltage gains, improved stability, and enhanced power density. These advancements combined with the integration of the grid technologies for real-life energy management and not only enhance performance and reliability but also reduce the dependence on fossil fuels and address global climate changes, particularly used in applications that require efficient power conversion. The motivations behind this in which inventions is to create a sustainable energy feature that can meet the globe growing global demand while minimizing the environmental impacts.

Methodology

The proposed converter enhances the ability to deliver continuous power more effectively by ensuring stability and the reliable flow of energy this method optimizes the conversion process and reduces the power interruptions and helps in enhancing the efficacy of the whole system by maintaining the consistent current it minimizes the fluctuations that can lead to the inefficiencies and making it particularly beneficial for the applications that requires the continuous power supply. In contrast, it is essential to keep in mind that real-time scenarios often experience time delays in control unit, and result in either poor control performance or divergence of the system states. Thus it seems more complicate to handle the conventional controllers in these scenarios. Due to the fact that, the traditional Smith predictor (SP) eliminates the input delay from the control unit to remove its influences on the system output, this research could be possibly lead to numerous discussions as given through the works21,22,23,24,25,26. Also, the traditional SP approach could not be effective in the case of exogenous disturbances, system modeling faults, and random varying delays. In order to handle this situation, the modified Smith predictor (MSP) technique integrated with disturbance estimator block has been discussed in27,28. Even, the integrated set up improves the disturbance elimination, the filtering issue of high frequency signals on the estimation time line was not taken care in the above said research works. Also, the above discussed methods are conservative, not suitable, while handling with unstable time delay systems with measurement noise. In such cases, the controller design becomes more challenging and motivates us to incorporate a filter together with the traditional Smith predictor’s primary feedback path, which enhances the tuning challenges as discussed through29,30,31,32. This incorporation neutralizes the crucial system action and removes the influences of the input time lag, which varies regarding time. In particular, the disturbance estimator involving in the existing works31,32are HEID and IEID respectively. Further, the disturbance-prediction-based control approaches are discussed in the works33,34, where the disturbance observer and EID framework have been designed for disturbance elimination, individually. The IEID method provides disturbance compensation in the works35,36for complex dynamical systems and cyber-physical systems, respectively. Based on the above discussed works, it could be noted that the disturbance observer designed in the work33needs the prior information about the disturbances while EID based methods do not require the details of the disturbances and yield better disturbance estimation and reduction in the matched disturbance case. But, it is unavoidable and unendurable the entrance of external disturbances through any input channel of the control systems in real-time scenario. Though, EID approach is an effective active disturbance reduction method, it is not suitable in the case of mismatched disturbances. However, the disturbance observer designed in the work33 uses prediction-based control technique and provides good disturbance reduction, it needs the information about the unknown disturbances, which are often unavailable. Thus, among the above discussed active disturbance reduction techniques, there is a research gap and need of novel disturbance estimator which does not require the details about the disturbances and should compensate both matched and mismatched disturbances. This gap could be fulfilled in our proposed work, by the integration of enhanced disturbance estimator with traditional SP block, which provides better disturbance elimination in case of both matched and mismatched disturbances without knowing their complete information. Thus, the integration of enhanced disturbance estimator with traditional SP and filter is known as enhanced SP (ESP) approach, which motivates to enhance the elimination of both disturbance and delay effects on the system output.

In addition to the above-described advanced topics, tracking control has become a popular and important topic in current research on control systems due to its effectiveness in various real-time applications. In reality, tracking control is commonly used to follow the required command by the output of the considered control environment subject to unknown disturbances, delays, nonlinearities and so on. The main intension of tracking control is to compel the output or the states of the control system to follow the desired specific reference input. This version highlights the importance of tracking control in academic and practical study of real-world scenarios. In particular, repetitive control is one among the famous tracking controller design introduced in 1990s by the Japanese researchers. On the basis of internal model concept, the repetitive control (RC) technique has the attractive design to enhance the tracking accuracy by removing the periodic external disturbances. Due to its efficacy and simple construction, RC technique has gradual development and most of the tracking controller works have been accomplished by using RC technique. The traditional RC technique fails in the case of control systems having relative degree zero. In such cases, the delay line of RC structure is connected with a low pass filter to achieve an accurate tracking performance, then the resulted framework is known as modified RC (MRC) which has numerous applications in the control systems research (for example see37,38,39,40,41,42,43,44and the cited works therein.) In particular, MRC approach for various type of dynamical systems has been discussed in37,38,39,40, in which the periodic disturbances elimination and periodic reference tracking could be achieved without steady-state error. Also, the MRC technique for electrical vehicle charger system has presented the efficiency of repetitive learning strategy in41. Additionally, the adaptive repetitive control algorithm with event-triggering has been proposed for the nonlinear systems in42, where the MRC design for fractional order systems in43,44 have provided the precise tracking controller design. Thereby, expanding the conversions from the previous sections, the investigation enlarged to enhance the MRC based structure for various type of dynamical systems subject to input time delays and disturbances. These developments not only broaden our theoretical results for the delayed nonlinear double boost converter in micro grid systems but also facilitate their practical applications, which emphasize their necessity in the extended area of control theory.

Novelty

The double boost converter used in the proposed PV based microgrid system requires DC voltage control for several key reasons such as Maximum Power Point Tracking (MPPT), voltage regulation for grid or battery interface, stability and protection, efficiency optimization, DC link stability in inverters and so on. It ensures that the PV panel operates at its MPPT, stabilizes the DC grid, manages power flow between generation, storage, and loads, and protects system components from over voltage or under voltage conditions. This double boost converter in microgrid environments has been affected by the input voltage variations, load variations, grid voltage variation, switching noise, control loop delays and so on. However, addressing these issues is crucial for managing the reliability and efficacy of double boost converters in microgrid environments. The innovations of current research on double boost converters in microgrid environments are listed below:

  • For mitigating the disturbance effects in the considered double boost converter systems, the disturbance estimator in ESP block has been designed.

  • The control loop delay effects in double boost converter systems are rectified by the appropriate construction of the filtered SP structure in ESP block.

  • The voltage regulation will be more essential in power systems due to the enhancing popularity of microgrids and distributed energy resources, which motivates us to construct the suitable MRC framework to handle the output voltage regulation in double boost converter.

  • In accordance with above said disturbance effects, the input voltage fluctuations are considered as matched disturbances, which could be handled effectively by the existing EID and SP based method23. But the existing EID and SP based method has limits in providing accurate disturbance estimation in case of mismatched disturbances. However, the proposed ESP based MRC technique gives better disturbance mitigation during mismatched disturbance scenarios like load variations, grid voltage variation, switching noise and some others.

Contribution

Based on the previous discussions, the important objective of the proposed study is to ensure the tracking controller technique by using the ESP based MRC structure for a time delayed nonlinear double boost converter influenced by input time lags and exogenous disturbances. An outline of the major components and contributions towards this study is presented below;

  1. (i)

    Applying the mathematical model to the nonlinear double boost converter with input delays and external disturbances, the nonlinear control system has been designed to regulate the output voltage by using MRC technique.

  2. (ii)

    Consequently, by combining Lyapunov-Krasovskii functional with LMIs, the required conditions are formed in a group of matrix inequalities which guarantee the asymptotic stability of the suggested nonlinear double boost converter system. The controller parameters are obtained by solving these LMIs.

  3. (iii)

    The theoretical results are validated through the simulation analysis, which clarifies the effectiveness of the ESP based MRC technique over the existing methods like HEID, PEID and IEID methods discussed through45,46,47, accordingly.

A framework of the remaining part of this work is organized as in the subsequent: The nonlinear double boost converter system is modeled in Section. The associated controller design for the considered nonlinear double boost converter system has been provided in Section. The derivation of LMIs in Section guarantee the asymptotic stability condition of the nonlinear double boost converter model under consideration. Simulation analysis and comparison results are demonstrated in Section to strengthen the obtained theoretical discussions. Enduringly, Section includes the conclusion part of this research.

System description-overview of double boost converter

Fig. 2
figure 2

Overall Circuit Diagram of Double Boost Converter.

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Considering the state variables as inductor current \(I_{L_1}\), \(I_{L_2}\) and the output capacitor voltage \(V_C\) from the overall circuit diagram as displayed in Figure 2. The state space equation for different modes of the considered double boost converter circuit system is defined in this section. This operation of the double boost converter is explained in three modes of operation Mode-1, Mode-2 and Mode-3.

Mode-1: This mode starts when \(S_1\) is switched on and the inductors \(L_1\) and \(L_2\) are charging through the switch \(S_1\) as shown in the Figure 3. Thus, the KVL equations governs this circuit in Mode-1 are given in the following equations (1)-(4):

Fig. 3
figure 3

Mode-1.

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$$\begin{aligned}&\begin{array}{rl} V_g& =I_{L_2}r_2+L\frac{\textrm{d}I_{L_2}}{\textrm{d}t},\\ \frac{\textrm{d}I_{L_2}}{\textrm{d}t}& =\frac{V_g}{L_2}-\frac{I_{L_2}r_2}{L_2},\\ \dot{x}_2& =\frac{V_g}{L_2}-\frac{I_{L_2}r_2}{L_2}. \end{array} \end{aligned}$$
(1)
$$\begin{aligned}&\begin{array}{rl} & V_\mathbb {C}+\mathbb {C}r_\mathbb {C}\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t}-(I_{L_1} -\mathbb {C}\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t})\mathbb {R} =0, \\ & \quad \frac{RI_{L_1}}{(r_\mathbb {C}+\mathbb {R})}-\frac{V_\mathbb {C}}{\mathbb {C}(r_\mathbb {C} +\mathbb {R})}=\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t}, \\ & \quad \frac{RI_{L_1}}{(r_\mathbb {C}+\mathbb {R})}-\frac{V_\mathbb {C}}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})}=\dot{x}_3 \end{array} \end{aligned}$$
(2)
$$\begin{aligned}&\begin{array}{rl} & V_g-I_{L_1}r_1-L_1\frac{\textrm{d}I_{L_1}}{\textrm{d}t}-\mathbb {R}I_{L_1}+\mathbb {R} \mathbb {C}\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t} =0,\\ & \quad V_g-I_{L_1}r_1-L_1\frac{\textrm{d}I_{L_1}}{\textrm{d}t}-\mathbb {R}I_{L_1}+\mathbb {R} \mathbb {C}(\frac{RI_{L_1}}{(r_\mathbb {C}+\mathbb {R})}-\frac{V_\mathbb {C}}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})})= 0,\\ & \quad \left( \frac{-r_1r_\mathbb {C}-r_1\mathbb {R}-\mathbb {R}r_\mathbb {C}}{L_1(r_\mathbb {C} +\mathbb {R})}\right) I_{L_1}-\frac{\mathbb {R}V_\mathbb {C}}{L_1(r_\mathbb {C}+\mathbb {R})} +\frac{V_g}{L_1}=\frac{\textrm{d}I_{L_1}}{\textrm{d}t},\\ & \quad \left( \frac{-r_1r_\mathbb {C}-r_1\mathbb {R}-\mathbb {R}r_\mathbb {C}}{L_1(r_\mathbb {C} +\mathbb {R})}\right) I_{L_1}-\frac{\mathbb {R}V_\mathbb {C}}{L_1(r_\mathbb {C}+\mathbb {R})}+\frac{V_g}{L_1}=\dot{x}_1. \end{array} \end{aligned}$$
(3)
$$\begin{aligned}&\begin{array}{rl} & RI_{L_1}-\mathbb {R}\mathbb {C}\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t}=V_o,\\ & \quad RI_{L_1}-R\mathbb {C}(\frac{RI_{L_1}}{(r_\mathbb {C}+\mathbb {R})}+\frac{V_\mathbb {C}}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})})=V_o,\\ & \quad \frac{\mathbb {R}r_\mathbb {C}I_{L_1}}{(r_\mathbb {C}+\mathbb {R})}+\frac{\mathbb {R}V_\mathbb {C}}{(r_\mathbb {C}+\mathbb {R})} =V_o,\\ & \quad \begin{bmatrix} \frac{\mathbb {R}r_\mathbb {C}}{(r_\mathbb {C}+\mathbb {R})} & 0 & \frac{\mathbb {R}}{(r_\mathbb {C}+\mathbb {R})} \end{bmatrix} \begin{bmatrix} I_{L_1} \\ I_{L_2} \\ V_\mathbb {C} \end{bmatrix} =V_o. \end{array} \end{aligned}$$
(4)

It follows from (1)-(4) that the circuit model for Mode-1 is derived as

$$\begin{aligned} \begin{array}{lll} \dot{x} = A_1x + B_1u \\ y=E_1x+D_1u,\\ \end{array} \end{aligned}$$
(5)

where \(x=\begin{bmatrix} I_{L_1}&I_{L_2}&V_\mathbb {C} \end{bmatrix}^T\), \(u=V_g\) and \(y=V_o\). Also, the corresponding state matrices in (5) are computed

\(A_1=\begin{bmatrix} \frac{-r_1r_\mathbb {C}-r_1\mathbb {R}-\mathbb {R}r_\mathbb {C}}{L_1(r_\mathbb {C}+\mathbb {R)}} & 0 & \frac{-\mathbb {R}}{L_1(r_\mathbb {C}+\mathbb {R})}\\ 0 & -\frac{r_2}{L_2} & 0\\ \frac{\mathbb {R}}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})} & 0 & -\frac{1}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})} \end{bmatrix}\), \(B_1=\begin{bmatrix} \frac{1}{L_1} \\ \frac{1}{L_2} \\ 0 \end{bmatrix}\), \(E_1=\begin{bmatrix} \frac{\mathbb {R}r_\mathbb {C}}{(r_\mathbb {C}+\mathbb {R})}&0&\frac{\mathbb {R}}{(r_\mathbb {C}+\mathbb {R})} \end{bmatrix}\) and \(D_1=0\).

Mode-2: This operating mode starts when \(S_2\) is triggered on. The Mode-2 starts during both \(S_1\) and \(S_2\) are switched on and the inductors \(L_1\) and \(L_2\) are charging through the switch \(S_1\) and \(S_2\) as shown in the Figure 4. Thus, the KVL equations governs this circuit in Mode-2 are given from equation (6)-(8):

Fig. 4
figure 4

Mode-2.

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$$\begin{aligned}&\begin{array}{rl} \frac{\textrm{d}I_{L_1}}{\textrm{d}t}& =\frac{V_g}{L_1}-\frac{I_{L_1}r_1}{L_1},\\ \dot{x}_1& =\frac{V_g}{L_1}-\frac{I_{L_1}r_1}{L_1},\\ \frac{\textrm{d}I_{L_2}}{\textrm{d}t}& =\frac{V_g}{L_2}-\frac{I_{L_2}r_2}{L_2},\\ \dot{x}_2& =\frac{V_g}{L_2}-\frac{I_{L_2}r_2}{L_2}. \end{array} \end{aligned}$$
(6)
$$\begin{aligned}&\begin{array}{rl} & V_\mathbb {C}+\mathbb {C}r_\mathbb {C}\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t} -(\mathbb {C}\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t})\mathbb {R} =0, \\ & \quad \frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t}=-\frac{V_\mathbb {C}}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})}, \\ & \quad \dot{x}_3=-\frac{V_\mathbb {C}}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})}. \end{array} \end{aligned}$$
(7)
$$\begin{aligned}&\begin{array}{rl} & \mathbb {R}\mathbb {C}\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t}=V_o,\\ & \quad -\frac{\mathbb {R}V_\mathbb {C}}{(r_\mathbb {C}+\mathbb {R})} =V_o,\\ & \quad \begin{bmatrix} 0 & 0 & -\frac{\mathbb {R}}{(r_\mathbb {C}+\mathbb {R})} \end{bmatrix} \begin{bmatrix} I_{L_1} \\ I_{L_2} \\ V_\mathbb {C} \end{bmatrix} =V_o. \end{array} \end{aligned}$$
(8)

It obviously considered from (6)-(8) that the circuit model for Mode-2 is derived as

$$\begin{aligned} \begin{array}{lll} \dot{x} = A_2x + B_2u \\ y=E_2x+D_2u,\\ \end{array} \end{aligned}$$
(9)

where \(x=\begin{bmatrix} I_{L_1}&I_{L_2}&V_\mathbb {C} \end{bmatrix}^T\), \(u=V_g\) and \(y=V_o\). Also, the state matrices in the state space form (9) are computed as \(A_2=\begin{bmatrix} \frac{-r_1}{L_1} & 0 & 0\\ 0 & -\frac{r_2}{L_2} & 0\\ 0 & 0 & -\frac{1}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})} \end{bmatrix}\), \(B_2=\begin{bmatrix} \frac{1}{L_1} \\ \frac{1}{L_2} \\ 0 \end{bmatrix}\), \(E_2=\begin{bmatrix} 0&0&-\frac{\mathbb {R}}{(r_\mathbb {C}+\mathbb {R})} \end{bmatrix}\) and \(D_2=0\).

Mode-3: This operating Mode-3 starts when \(S_1\) is switched OFF and \(S_2\) is still ON as shown in the Figure 5. Thus, the KVL equations governs this circuit in Mode-3 are given in the equations (10)-(13):

Fig. 5
figure 5

Mode-3.

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$$\begin{aligned}&\begin{array}{rl} & V_g-I_{L_1}r_1-L_1\frac{\textrm{d}I_{L_1}}{\textrm{d}t} =0,\\ & \quad \frac{V_g}{L_1}-\frac{I_{L_1}r_1}{L_1}=\frac{\textrm{d}I_{L_1}}{\textrm{d}t},\\ & \quad \frac{V_g}{L_1}-\frac{I_{L_1}r_1}{L_1}=\dot{x}_1. \end{array} \end{aligned}$$
(10)
$$\begin{aligned}&\begin{array}{rl} & V_g-I_{L_2}r_2-L_2\frac{\textrm{d}I_{L_2}}{\textrm{d}t}-\mathbb {R}I_{L_2} +\mathbb {R}\mathbb {C}\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t} =0,\\ & \quad V_g-I_{L_2}r_2-L_2\frac{\textrm{d}I_{L_2}}{\textrm{d}t}-\mathbb {R}I_{L_2} +\mathbb {R}\mathbb {C}(\frac{RI_{L_2}}{(r_\mathbb {C}+\mathbb {R})}-\frac{V_\mathbb {C}}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})})= 0,\\ & \quad \left( \frac{-r_2r_\mathbb {C}-r_2\mathbb {R}-\mathbb {R}r_\mathbb {C}}{L_2(r_\mathbb {C} +\mathbb {R})}\right) I_{L_2}-\frac{\mathbb {R}V_\mathbb {C}}{L_2(r_\mathbb {C}+\mathbb {R})} +\frac{V_g}{L}=\frac{\textrm{d}I_{L_2}}{\textrm{d}t},\\ & \quad \left( \frac{-r_2r_\mathbb {C}-r_2\mathbb {R}-\mathbb {R}r_\mathbb {C}}{L_2(r_\mathbb {C} +\mathbb {R})}\right) I_{L_2}-\frac{\mathbb {R}V_\mathbb {C}}{L_2(r_\mathbb {C}+\mathbb {R})}+\frac{V_g}{L_2}=\dot{x}_2. \end{array} \end{aligned}$$
(11)
$$\begin{aligned}&\begin{array}{rl} & V_\mathbb {C}+\mathbb {C}r_\mathbb {C}\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t} -(I_{L_2}-\mathbb {C}\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t})\mathbb {R} =0, \\ & \quad \frac{RI_{L_2}}{(r_\mathbb {C}+\mathbb {R})}-\frac{V_\mathbb {C}}{\mathbb {C}(r_\mathbb {C} +\mathbb {R})}=\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t}, \\ & \quad \frac{RI_{L_2}}{(r_\mathbb {C}+\mathbb {R})}-\frac{V_\mathbb {C}}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})}=\dot{x}_3 \end{array} \end{aligned}$$
(12)
$$\begin{aligned}&\begin{array}{rl} & RI_{L_2}-\mathbb {R}\mathbb {C}\frac{\textrm{d}V_\mathbb {C}}{\textrm{d}t}=V_o,\\ & \quad RI_{L_2}-R\mathbb {C}(\frac{RI_{L_2}}{(r_\mathbb {C}+\mathbb {R})}+\frac{V_\mathbb {C}}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})})=V_o,\\ & \quad \frac{\mathbb {R}r_\mathbb {C}I_{L_2}}{(r_\mathbb {C}+\mathbb {R})}+\frac{\mathbb {R}V_\mathbb {C}}{(r_\mathbb {C}+\mathbb {R})} =V_o,\\ & \quad \begin{bmatrix} 0 & \frac{\mathbb {R}r_\mathbb {C}}{(r_\mathbb {C}+\mathbb {R})} & \frac{\mathbb {R}}{(r_\mathbb {C}+\mathbb {R})} \end{bmatrix} \begin{bmatrix} I_{L_1} \\ I_{L_2} \\ V_\mathbb {C} \end{bmatrix}=V_o. \end{array} \end{aligned}$$
(13)

It follows from (10)-(13) that the circuit model for Mode-3 is derived as

$$\begin{aligned} \begin{array}{lll} \dot{x} = A_3x + B_3u \\ y=E_3x+D_3u,\\ \end{array} \end{aligned}$$
(14)

where \(x=\begin{bmatrix} I_{L_1}&I_{L_2}&V_\mathbb {C} \end{bmatrix}^T\), \(u=V_g\) and \(y=V_o\). Also, the related state matrices in (14) are computed as

\(A_3=\begin{bmatrix} -\frac{r_1}{L_1} & 0 & 0\\ 0 & \frac{-r_2r_\mathbb {C}-r_2\mathbb {R}-\mathbb {R}r_\mathbb {C}}{L_2(r_\mathbb {C}+\mathbb {R)}} & \frac{\mathbb {R}}{L_2(r_\mathbb {C}+\mathbb {R})} \\ 0 & \frac{\mathbb {R}}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})} & -\frac{1}{\mathbb {C}(r_\mathbb {C}+\mathbb {R})} \end{bmatrix}\), \(B_3=\begin{bmatrix} \frac{1}{L_1} \\ \frac{1}{L_2} \\ 0 \end{bmatrix}\), \(E_3=\begin{bmatrix} 0&\frac{\mathbb {R}r_\mathbb {C}}{(r_\mathbb {C}+\mathbb {R})}&\frac{\mathbb {R}}{(r_\mathbb {C}+\mathbb {R})} \end{bmatrix}\) and \(D_3=0\).

Fig. 6
figure 6

Augmented Circuit Model.

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Augmented Circuit Model: One full cycle operation involves Mode-1, Mode-2 and Mode-3 for the given duty cycle. As shown in Figure 6, the Augmented Circuit Model is obtained by summing up the all above mentioned modes and given by (15)

$$\begin{aligned} \begin{array}{rl} \dot{x}(t) & = Ax(t) + Bu(t)\\ y(t)& =Cx(t) \end{array} \end{aligned}$$
(15)

where the state matrices are represented as \(A= A_1 d_1+0.2 A_2+A_3 d_2,\) \(B= B_1 d_1+0.2B_2+B_3 d_2,\) and \(C= E_1 d_1+0.2 E_2+E_3 d_2\). Thus, \(A=\begin{bmatrix} A_{(1,1)} & 0 & -\frac{\mathbb {R}d_1}{L_1(r_\mathbb {C}+\mathbb {R})}\\ 0 & A_{(2,2)} & \frac{\mathbb {R}d_2}{L_2(r_\mathbb {C}+\mathbb {R})} \\ \frac{\mathbb {R}d_1}{\mathbb {C}(r_c+\mathbb {R})} & \frac{\mathbb {R}d_2}{\mathbb {C}(r_c+\mathbb {R})} & \frac{-d_1-0.2-d_2}{\mathbb {C}(r_c+\mathbb {R})} \end{bmatrix}\), \(B=\begin{bmatrix} \frac{d_1+0.2+d_2}{L_1} \\ \frac{d_1+0.2+d_2}{L_2} \\ 0 \end{bmatrix}\), \(E=\begin{bmatrix} \frac{d_1\mathbb {R}r_\mathbb {C}}{(r_\mathbb {C}+\mathbb {R})} \\ \frac{d_2\mathbb {R}r_\mathbb {C}}{(r_\mathbb {C}+\mathbb {R})} \\ \frac{d_1\mathbb {R}-0.2\mathbb {R}+d_2\mathbb {R}}{(r_\mathbb {C}+\mathbb {R})} \end{bmatrix}^T,\)

where \(A_{(1,1)}=\frac{-r_1r_\mathbb {C}d_1-r_1\mathbb {R}d_1-r_{\mathbb {C}}\mathbb {R}d_1-0.2r_1r_ \mathbb {C}-0.2r_1\mathbb {R}-r_1r_\mathbb {C}d_2-r_1\mathbb {R}d_2}{L_1(r_\mathbb {C}+\mathbb {R})}\),

and \(A_{(2,2)}=\frac{-r_2r_\mathbb {C}d_1-r_2\mathbb {R}d_1-0.2 r_2r_\mathbb {C}-0.2 r_2\mathbb {R}+r_2r_\mathbb {C}d_2+r_2\mathbb {R}d_2+\mathbb {R}r_\mathbb {C}d_2}{L_1(r_c+\mathbb {R})}\).

Controller description

In this section, we derive the controller design for the double boost converter model (15) subject to state delay, saturation nonlinearity, input delay, nonlinear uncertainty and disturbances, accordingly. Now, the converter model (15) is extended as the following nonlinear time-delayed double boost converter dynamical systems,

$$\begin{aligned} \begin{array}{rl} \dot{x}(t)& =Ax(t)+A_h x(t-h(t))+B \varrho [u(t-\tau _u)]+N_u(t,x(t))+B_w w(t),\\ y(t)& =Cx(t), \end{array} \end{aligned}$$
(16)

where \(x(t) \in \mathscr {R}^x\) and \(y(t) \in \mathscr {R}^y\) define the state and output signals of the considered double boost converter system, respectively; \(\varrho [.]:\mathscr {R}^u \rightarrow \mathscr {R}^u\) denotes the nonlinear actuator saturation with \(\varrho [u(t)]=\left[ \varrho ^T(u_1(t)),\ldots ,\varrho ^T(u_u(t)))\right] ^T\); \(u(t-\tau _u) \in \mathscr {R}^u\) is the control channel having invariant time-delay \(\tau _u\) and h(t) denotes the non-differentiable varying delay with \(0 < h_m \le h(t) \le h_M\), \(h_m \ne h_M\); w(t) is the external disturbance input; \(N_u(t,x(t))\) describes the nonlinear function in the system; A, \(A_h\), B, \(B_{w}\), and C represent the state matrices of adaptable dimensions.

Fig. 7
figure 7

Structure of ESP based MRC framework for the delayed double boost converter.

Full size image

Further, the existence of time delays and disturbances in control input units can lead to the divergence and oscillations of the system states, which severely degrade the control performance and potentially lead to system destabilization. As shown in Figure 7, which illustrates that the layout of ESP based MRC technique for the time-delayed double boost converter is combined into the proposed disturbance estimator block to handle the issues of delay and disturbances. The overall systematic framework covers five primary sub-segments: the time delayed boost converter system, the disturbance estimator \(D_\gamma (s)\), the nominal time delayed double boost converter system, MRC block, and filter \(F_\beta (s)\) in ESP block. The system successfully achieves the desired tracking accuracy, disturbance elimination, and delay effect rectification through the proper framework of ESP based MRC technique. Thus, in order to force the output of the double boost converter system to track the provided reference input \(r_y(t)\), MRC techniques integrated with ESP have to be implemented. By rejecting periodic disturbances, the internal model principle is the basis upon which the MRC is built for allowing the system output to closely match the desired reference. The MRC block contains the low pass filter \(M_c(s)=\frac{\omega _c}{s+\omega _c}\), where \(\omega _c\) represents the cut off angular frequency of the reference input. Then, the compact state space form of MRC framework is

$$\begin{aligned} \begin{array}{lll} \dot{x}_{c}(t) = -\omega _cx_c(t)+\omega _cx_c(t-P_e)+\omega _c e_y(t)\\ \end{array} \end{aligned}$$
(17)

where \(e_y(t)=r_y(t)-y_{\alpha }(t)-y_{\beta }(t)\) is the tracking error and \(P_e\) is the reference period; \(y_\alpha (t)\) is the output of the nominal system with saturation; \(y_\beta (t)\) describes the output of the filter in ESP block. The nominal system has been written as

$$\begin{aligned} \begin{array}{rll} \dot{x}_\alpha (t)& =& A_\alpha x_\alpha (t)+A_{h\alpha } x_\alpha (t-h(t))+B_\alpha \varrho \left[ u_\alpha (t)\right] ,\\ y_{\alpha }(t)& =& C_\alpha x_\alpha (t), \end{array} \end{aligned}$$
(18)

where \(x_\alpha (t) \in \mathscr {R}^x\), \(u_\alpha (t)\in \mathscr {R}^u\) and \(y_\alpha (t) \in \mathscr {R}^y\) define the state, control and output vectors of the nominal double boost converter system, respectively. In addition, \(A_{\alpha }\), \(A_{h\alpha }\), \(B_{\alpha }\), and \(C_{\alpha }\) are the appropriate dimension matrices. Due to the fact that the integration of filter in traditional SP block improves the system stability by removing the effect of high-frequency components and model uncertainties which leads system instability. The state space form of the filter \(F_\beta (s)\) in ESP block is

$$\begin{aligned} \begin{array}{rll} \dot{x}_\beta (t)& =& A_\beta x_\beta (t)+B_\beta \left[ y(t)-y_\alpha (t-\tau _u)\right] ,\\ y_{\beta }(t)& =& C_\beta x_\beta (t), \end{array} \end{aligned}$$
(19)

where, \(x_\beta (t)\) and \(y_\beta (t)\) define the state and output signals of \(F_\beta (s)\), accordingly; \(A_{\beta }\), \(B_{\beta }\), and \(C_{\beta }\) are the suitable constant matrices. Additionally, the disturbances can affect the system stability or control performance, the traditional SP may not sufficient to eliminate the disturbance effects in the system output. This issue is handled by the disturbance estimator \(D_\gamma (s)\) incorporated with traditional SP block, the resulted structure is known as ESP block. The associated state-matrix framework of the disturbance estimator \(D_\gamma (s)\) is provided with the subsequent compact form;

$$\begin{aligned} \begin{array}{rll} \dot{x}_\gamma (t)& =& A_\gamma x_\gamma (t)+B_\gamma \left[ y(t)-y_\alpha (t-\tau _u)\right] ,\\ y_{\gamma }(t)& =& B^+B_wC_\gamma x_\gamma (t)+B^+B_wD_\gamma \left[ y(t)-y_\alpha (t-\tau _u)\right] , \end{array} \end{aligned}$$
(20)

where, \(x_\gamma (t)\) and \(y_\gamma (t)\) denote the state and output signals of the \(D_\gamma (s)\), accordingly; \(A_{\gamma }\), \(B_{\gamma }\), \(C_{\gamma }\), and \(D_{\gamma }\) are the adaptable dimension matrices and \(B^+\) is the generalized inverse of the matrix B. Next, the enhanced control input has been computed as

$$\begin{aligned} \begin{array}{rll} u(t-\tau _u)& =& u_\alpha (t-\tau _u)-y_\gamma (t),\\ u_\alpha (t) & =& K_\alpha x_\alpha (t)+K_cx_c(t), \end{array} \end{aligned}$$
(21)

At this point, it could be highlighted that the compensation for external disturbances after an input delay poses challenges in reactive control methods. In order handle this issue, as discussed in the works33,34,35,36 the predictor-based control methods could be designed which enables the anticipation of delays and lead the system for compensating disturbance signals pro-actively. Thus, in the proposed ESP-based MRC framework, integration of disturbance compensation after taking input delay into account is technically feasible and enables for maintaining the required control performance and system stability.

Let \(K_\alpha\), \(K_c\), \(G_\alpha\), \(G_c\) \(\in \mathscr {R}^{u \times u}\) and \(x_\alpha (t)\), \(x_c(t)\) \(\in \Psi (t)\), then it follows from the Lemma 2 of48 that

$$\varrho \left[ u_\alpha (t)\right] =\sum \limits _{s=1}^{2^u} \phi _s \left[ (\tilde{K}_\alpha )x_\alpha (t)+(\tilde{K}_c)x_c(t)\right] ,$$

where \(s\in\) sat\(=\{1,2,\ldots ,2^u\}\), \(\phi _s\) denotes the scalars which satisfy \(\sum \limits _{s=1}^{2^u}\phi _s=1\) and \(\phi _s\in [0,1]\), \(\tilde{K}_\alpha = (K_\alpha +G_\alpha )\) and \(\tilde{K}_c= (K_c+G_c)\) Now, the new state for the expanded closed-loop system has the form \(\zeta (t)=\begin{bmatrix} x^T(t)&x_c^T(t)&x_\alpha ^T(t)&x_\beta ^T(t)&x_\gamma ^T(t) \end{bmatrix}^T.\) Next, it could be underlined that the exterior signals have no influence on the design of proposed tracking and disturbance elimination control technique or the stability of the upgraded system, which leads to consider \(r_y(t)=0\), \(N_u(t,x(t))=0\) and \(w(t)=0\) in solving the important results. Henceforth, the considered double boost converter system, MRC structure, nominal double boost converter system, and \(F_\beta (s)\), \(D_\gamma (s)\), could be rewritten as follows;

$$\begin{aligned} \begin{array}{rll} \dot{x}(t)& =& Ax(t)+A_h x(t-h(t))+B\left\{ \sum \limits _{s=1}^{2^u} \phi _s\left[ \tilde{K}_\alpha x(t-\tau _u)+\tilde{K}_cx_c(t-\tau _u)\right] \right\} \\ & & -BC_\gamma x_\gamma (t)-BD_\gamma y(t)+ BD_\gamma y_\alpha (t-\tau _u),\\ \dot{x}_{c}(t) & =& -\omega _cx_c(t)+\omega _cx_c(t-P_e)-\omega _c y_\alpha (t)-\omega _c y_\beta (t)\\ \dot{x}_\alpha (t)& =& A_\alpha x_\alpha (t)+A_{h\alpha } x_\alpha (t-h(t)) +B_\alpha \left\{ \sum \limits _{s=1}^{2^u} \phi _s \left[ \tilde{K}_\alpha x(t)+\tilde{K}_cx_c(t)\right] \right\} ,\\ \dot{x}_\beta (t)& =& A_\beta x_\beta (t)+B_\beta \left[ y(t)-y_\alpha (t-\tau _u)\right] ,\\ \dot{x}_\gamma (t)& =& A_\gamma x_\gamma (t)+B_\gamma \left[ y(t)-y_\alpha (t-\tau _u)\right] . \end{array} \end{aligned}$$
(22)

Thus, the above discussions provides the following upgraded closed-loop system for further research,

$$\begin{aligned} \dot{\zeta }(t)=\sum \limits _{s=1}^{2^u}\phi _s\hat{\mathscr {A}}\zeta (t) +\hat{\mathscr {A}}_h\zeta (t-h(t))+\sum \limits _{s=1}^{2^u} \phi _s\hat{\mathscr {B}}_1\zeta (t-\tau _u)+\hat{\mathscr {B}}_2\zeta (t-P_e), \end{aligned}$$
(23)

where \(\hat{\mathscr {A}} =\left[ \begin{array}{ccccc} A-BB^+B_w D_\gamma C & 0 & 0 & 0 & -B B^+B_wC_\gamma \\ 0 & -\omega _cI & -\omega _cC_\alpha & -\omega _cC_\beta & 0 \\ 0 & B_\alpha \tilde{K}_c & A_\alpha -B_\alpha \tilde{K}_\alpha & 0 & 0 \\ B_\beta C & 0 & 0 & A_\beta & 0 \\ B_\gamma C & 0 & 0 & 0 & A_\gamma \end{array}\right]\),

\(\hat{\mathscr {A}}_h=\text{ diag }\{A_h, 0, A_{h\alpha }, 0, 0\}\), \(\hat{\mathscr {B}}_1 =\begin{bmatrix} 0 & B\tilde{K}_c & \begin{array}{c}BB^+B_w D_\gamma C_\alpha \\ -B\tilde{K}_\alpha \end{array} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -B_\beta C_\alpha & 0 & 0 \\ 0 & 0 & -B_\beta C_\alpha & 0 & 0 \end{bmatrix}\), and \(\hat{\mathscr {B}}_2=\text{ diag }\{0, -\omega _c I, 0, 0, 0\}\).

Results

This section will provide the desired ESP based MRC technique that will tackle the issues resulted by both delays and disturbances and achieve the required tracking performance for double boost converter systems. Particularly, a set of stability conditions are computed, which guarantee the asymptotic stability of the upgraded system (23). The consequent sufficient constraints are specifically obtained in LMI structure which could be solved with the help of Matlab LMI toolbox. It could be noted that the design of control parameters for the desired ESP based MRC technique are calculated simultaneously while solving the derived LMIs in Matlab LMI toolbox.

Theorem 0.1

For the known scalars \(h_m>0\), \(h_M>0\), \(\epsilon _1>0\), \(\epsilon _2>0\), \(\nu _b>0\) \((b=1,2,\ldots ,6)\) and \(\tau _u>0\), the upgraded system (23) is asymptotically stablizable, supposing that the existence of positive definite and symmetric matrices X, Y, \(\bar{Q}_x\), \(\tilde{Q}_x\), \(\hat{Q}_x\), \(R_x\), and \(\hat{\tilde{X}}_{4\alpha _i}\) and flexible dimension matrices \(L_{hx}\) \((h=1,2,3,4,5)\) leads to following LMIs hold:

$$\begin{aligned}&\begin{bmatrix} S_x & L_{1x} & L_{2x}\\ * & L_{3x} & L_{4x}\\ * & * & L_{5x} \end{bmatrix}>0, \end{aligned}$$
(24)
$$\begin{aligned}&R_x<\epsilon _2 X, \end{aligned}$$
(25)
$$\begin{aligned}&-S^{-1}<-\frac{Y}{\epsilon _1}, \end{aligned}$$
(26)
$$\begin{aligned}&\begin{bmatrix} \sqcup & \sqrt{h_{mM}} \Cap _1^{T} \\ * & -\frac{Y}{\epsilon _1} \end{bmatrix}<0, \end{aligned}$$
(27)

where \(\left[ \tilde{\sqcup }\right] _{6\times 6}=\hat{\tilde{\Lambda }}^x+h_{mM} \Cap _1^T S \Cap _1+\hat{\tilde{\sqcup }},\) with \(\hat{\tilde{\sqcup }}=\) with non-zero elements \(\hat{\tilde{\sqcup }}_{1,1}=\sum \limits _{s=1}^{2^u}\phi _s\hat{\mathscr {A}}X +\sum \limits _{s=1}^{2^u}\phi _sX\hat{\mathscr {A}}^T+\bar{Q}_x+\hat{Q}_x+\epsilon _2 X\), \(\hat{\tilde{\sqcup }}_{1,3}=\hat{\mathscr {A}}_{h}X\), \(\hat{\tilde{\sqcup }}_{1,5}=\sum \limits _{s=1}^{2^u}\phi _s\hat{\mathscr {B}}_{1}X\), \(\hat{\tilde{\sqcup }}_{1,6}=\hat{\mathscr {B}}_2Y\), \(\hat{\tilde{\sqcup }}_{2,2}=\tilde{Q}_x-\bar{Q}_x\), \(\hat{\tilde{\sqcup }}_{4,4}=-\tilde{Q}_x\), \(\hat{\tilde{\sqcup }}_{5,5}=-\hat{Q}_x\),

\(\hat{\tilde{\sqcup }}_{6,6}=-Y,\) \(\Cap _1=\begin{bmatrix}\hat{\mathscr {A}}X&0&\hat{\mathscr {A}}_{h}X&0&\hat{\mathscr {B}}_1X&\hat{\mathscr {B}}_2Y\end{bmatrix}\) and \(\hat{\tilde{\Lambda }}^x=\text{ diag }\{0,\Lambda _1^x+(h_{mM})\Lambda _2^x,0,0\}\),

where, \(X=\text{ diag }\{\nu _1X_1,\nu _2X_2,\nu _3X_3,\nu _4X_4,\nu _5X_5\}\), \(Y=\text{ diag }\{Y_1,\nu _6Y_2,Y_3,Y_4,Y_5\}\), \(S_x=\text{ diag }\{S_{x1},S_{x2},S_{x2},S_{x2},S_{x2}\}\),

\(L_{hx}=\text{ diag }\{L_{xh1},L_{xh2},L_{xh3},L_{xh4},L_{xh5}\}\), \((h=1,2,3,4,5)\), \(\tilde{Q}_x=\text{ diag }\{\tilde{Q}_{x1},\tilde{Q}_{x2},\tilde{Q}_{x3},\tilde{Q}_{x4},\tilde{Q}_{x5}\}\), \(\bar{Q}_x=\text{ diag }\{\bar{Q}_{x1},\bar{Q}_{x2},\bar{Q}_{x3},\bar{Q}_{x4},\bar{Q}_{x5}\}\), \(\hat{Q}_x=\text{ diag }\{\hat{Q}_{x1},\hat{Q}_{x2},\hat{Q}_{x3},\hat{Q}_{x4},\hat{Q}_{x5}\}\), \(R_x=\text{ diag }\{R_{x1},R_{x2},R_{x3},R_{x4},R_{x5}\}\) and \(S^{-1}=\text{ diag }\{S^{-1}_{1},S^{-1}_{2},S^{-1}_{3},S^{-1}_{4},S^{-1}_{5}\}\).

Also, \(\hat{\mathscr {A}}X =\left[ \begin{array}{ccccc} \nu _1AX_1-\nu _1B D_\gamma CX_1 & 0 & 0 & 0 & -\nu _5B C_\gamma X_5 \\ 0 & -\nu _2\omega _cX_2 & -\nu _3\omega _cC_\alpha X_3 & -\nu _4\omega _cC_\beta X_4 & 0 \\ & & & & \\ 0 & \nu _2B_\alpha (W_{kc}+W_{gc}) & \begin{array}{c}\nu _3A_\alpha X_3\\ -\nu _3B_\alpha W_{k\alpha }\\ -\nu _3B_\alpha W_{g\alpha } \end{array} & 0 & 0 \\ & & & & \\ \nu _1B_\beta C X_1 & 0 & 0 & \nu _4A_\beta X_4 & 0 \\ \nu _1 B_\gamma C X_1 & 0 & 0 & 0 & \nu _5 A_\gamma X_5 \end{array}\right]\),

\(\hat{\mathscr {A}}_hX=\text{ diag }\{\nu _1A_hX_1, 0, \nu _3A_{h\alpha }X_3, 0, 0\}\), \(\hat{\mathscr {B}}_2Y=\text{ diag }\{0, -\nu _6\omega _c Y, 0, 0, 0\}\), and

\(\hat{\mathscr {B}}_1 =\begin{bmatrix} 0 & \nu _2B_\alpha (W_{kc}+W_{gc}) & \nu _3B D_\gamma C_\alpha X_3-\nu _3B_\alpha (W_{k\alpha }+W_{g\alpha }) & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & -\nu _3B_\beta C_\alpha X_3 & 0 & 0 \\ 0 & 0 & -\nu _3B_\beta C_\alpha X_3 & 0 & 0 \end{bmatrix}\). Moreover, suppose that the above said LMIs (24)-(27) are feasible, then the saturation control values are computed with the relation,

$$\begin{aligned} K_\alpha =W_{k\alpha }X_3^{-1}, G_\alpha =W_{g\alpha }X_3^{-1}, K_c=W_{kc}X_2^{-1}, \text{ and } G_c=W_{gc}X_2^{-1}. \end{aligned}$$
(28)

Proof

Then, for deriving the asymptotic stability of upgraded closed-loop system (23), it is necessary to choose the Lyapunov-Krasovskii-Functional (LKF) as in the subsequent form,

$$\begin{aligned} \begin{array}{ll} V(\zeta (t),t)=& \zeta ^T(t)\mathscr {P} \zeta (t)+\int \limits _{t-h_m}^{t}\zeta ^T(a)\bar{Q}\zeta (a) \textrm{d}a+\int \limits _{t-h_M}^{t-h_m}\zeta ^T(a)\tilde{Q}\zeta (a) \textrm{d}a+\int \limits _{t-\tau _u}^{t}\zeta ^T(a)\hat{Q}\zeta (a)\textrm{d}a\\ & +\int \limits _{t-P_e}^{t}\zeta ^T(a)R\zeta (a)\textrm{d}a, +\int \limits _{-h_M}^{-h_m}\int \limits _{t+a_2}^{t} \ \dot{\zeta }^T(a_1)S \dot{\zeta }(a_1)\textrm{d}a_1\textrm{d}a_2, \end{array} \end{aligned}$$
(29)

wherein the matrices \(\mathscr {P}=\) diag\(\{\frac{1}{\nu _1}\mathscr {P}_{1}, \frac{1}{\nu _2}\mathscr {P}_{2},\frac{1}{\nu _3}\mathscr {P}_{3},\frac{1}{\nu _4} \mathscr {P}_{4},\frac{1}{\nu _5}\mathscr {P}_5\}\), \(\bar{Q}=\) diag\(\{\bar{Q}_{1},\bar{Q}_{2},\bar{Q}_{3},\bar{Q}_{4},\bar{Q}_{5}\}\), \(\tilde{Q}=\) diag\(\{\tilde{Q}_{1},\tilde{Q}_{2},\tilde{Q}_{3},\tilde{Q}_{4},\tilde{Q}_{5}\}\), \(\hat{Q}=\) diag\(\{\hat{Q}_{1},\hat{Q}_{2},\hat{Q}_{3},\hat{Q}_{4},\hat{Q}_{5}\}\), \(R=\) diag\(\{R_{1},\frac{1}{\nu _6}R_{2},R_{3},R_4,R_{5}\}\) and \(S=\) diag\(\{S_1,S_2,S_3,S_4,S_5\}\) satisfy the symmetric positive definite conditions. Further, taking the time derivative of the LKF along the solutions \(\zeta (t)\) provides as

$$\begin{aligned} \begin{array}{ll} \dot{V}(\zeta )=& 2\zeta ^T(t)\mathscr {P}\left[ \sum \limits _{s=1}^{2^u}\phi _s\hat{\mathscr {A}}\zeta (t) +\hat{\mathscr {A}}_h\zeta (t-h(t))+\sum \limits _{s=1}^{2^u}\phi _s\hat{\mathscr {B}}_1 \zeta (t-\tau _u)+\hat{\mathscr {B}}_2\zeta (t-P_e)\right] \\ & + \zeta ^T(t)(\bar{Q}+\hat{Q}+R)\zeta (t)+\zeta ^T(t-h_m)\tilde{Q}\zeta ^T(t-h_m)-\zeta ^T(t-h_m)\bar{Q} \zeta ^T(t-h_m)\\ & -\zeta ^T(t-h_M)\tilde{Q} \zeta ^T(t-h_M)-\zeta ^T(t-\tau _u)\hat{Q} \zeta ^T(t-\tau _u)-\zeta ^T(t-P_e)R \zeta ^T(t-P_e)\\ & +h_{mM}\dot{\zeta }^T(t)S\dot{\zeta }(t) -\int \limits _{t-h_M}^{t-h_m}\ \dot{\zeta }^T(a)S\dot{\zeta }(a)\textrm{d}a. \end{array} \end{aligned}$$
(30)

Let \(\hat{\Phi }(t)=[\zeta ^T(t),\zeta ^T(t-h_m),\zeta ^T(t-h(t)),\zeta ^T(t-h_M), \zeta ^T(t-\tau _u), \zeta ^T(t-P_e)]^T\). Next, for any suitable dimension matrices \(L_1, L_2, L_3, L_4, L_5\) which satisfy \(\begin{bmatrix} S & L_1 & L_2\\ * & L_3 & L_4\\ * & * & L_5 \end{bmatrix}>0\), by following the inequality in Theorem 3.3 of49, now the integral term in (30) can satisfy the subsequent condition,

$$\begin{aligned} -\int \limits _{t-h_M}^{t-h_m}\ \dot{\zeta }^T(a)S\dot{\zeta }(a)\textrm{d}a \le \Phi _1^T(t)\left[ \Lambda _1+(h_{mM})\Lambda _2\right] \Phi _1(t), \end{aligned}$$
(31)

where \(\Phi _1(t)=\left[ \zeta ^T(t-h_m) \ \zeta ^T(t-h(t))\ \zeta ^T(t-h_M)\right] ^T\),

$$\Lambda _1=\begin{bmatrix} L_1+L_1^T & -L_1^T+L_2 & 0\\ * & L_1+L_1^T-L_2-L_2^T & -L_1^T+L_2\\ * & * & -L_2-L_2^T \end{bmatrix},\ \Lambda _2=\begin{bmatrix} L_3 & L_4 & 0\\ * & L_3+L_5 & L_4\\ * & * & L_5 \end{bmatrix},\ h_{mM}=h_M-h_m.$$

It could be obtained from (30)-(31) that

$$\begin{aligned} \begin{array}{rl} \dot{V}(\zeta (t),t) \le&\hat{\Phi }^T(t) \left[ \tilde{\cup }\right] \hat{\Phi }(t) \end{array}, \end{aligned}$$
(32)

where \(\left[ \tilde{\cup }\right] _{6\times 6}=\hat{\tilde{\Lambda }}+h_{mM} \Cap _2^T S \Cap _2+\hat{\tilde{\cup }},\) with \(\hat{\tilde{\cup }}=\) with non-zero elements \(\hat{\tilde{\cup }}_{1,1}=2\mathscr {P}\sum \limits _{s=1}^{2^u}\phi _s\hat{\mathscr {A}}+\bar{Q}+\hat{Q}+R\), \(\hat{\tilde{\cup }}_{1,3}=\mathscr {P}\hat{\mathscr {A}}_{h}\), \(\hat{\tilde{\cup }}_{1,5}=\sum \limits _{s=1}^{2^u}\phi _s\mathscr {P}\hat{\mathscr {B}}_{1}\), \(\hat{\tilde{\cup }}_{1,6}=\mathscr {P}\hat{\mathscr {B}}_2\), \(\hat{\tilde{\cup }}_{2,2}=\tilde{Q}-\bar{Q}\), \(\hat{\tilde{\cup }}_{4,4}=-\tilde{Q}\), \(\hat{\tilde{\cup }}_{5,5}=-\hat{Q}\), \(\hat{\tilde{\cup }}_{6,6}=-R,\)

\(\Cap _2=\begin{bmatrix}\hat{\mathscr {A}}&0&\hat{\mathscr {A}}_{h}&0&\hat{\mathscr {B}}_1&\hat{\mathscr {B}}_2\end{bmatrix}\) and \(\hat{\tilde{\Lambda }}=\text{ diag }\{0,\Lambda _1+(h_{mM})\Lambda _2,0,0\}\). Let \(\mathscr {P}^{-1}=X\), \(R^{-1}=Y\), \(\bar{Q}_x=X\bar{Q}X\), \(R_x=XRX\), \(\tilde{Q}_x=X\tilde{Q}X\), \(\hat{Q}_x=X\hat{Q}X\), \(L_{hx}=XL_hX\), \((h=1,2,3,4,5)\). Further, consider \(S<\epsilon _1 R\), \(R<\epsilon _2\mathscr {P}\), \(W_{k\alpha }=K_\alpha X_3\), \(W_{g\alpha }=G_\alpha X_3\), \(W_{kc}=K_cX_2\), and \(W_{gc}=G_cX_2\). Next, by applying the pre- and post- multiplication for the inequalities \(\begin{bmatrix} S & L_1 & L_2\\ * & L_3 & L_4\\ * & * & L_5 \end{bmatrix}>0\) and \(R<\epsilon _2\mathscr {P}\) by X and X, individually, the inequality conditions (24) and (25) are hold. By taking inverse on both sides of \(S<\epsilon _1 R\), then the condition (26) is satisfied. Also, taking Schur complement and pre- and post- multiplying the inequality (32) by diag\(\{X,X,X,X,X, Y,I\}\), the condition (27) is satisfied. Thus, it could be clearly concluded from (32) and the LMIs (24)-(27) that \(\tilde{\cup }<0\) solely in the event that \(\dot{V}(\zeta (t),t)<0\) results that the upgraded closed-loop system (23) is asymptotically stable. Henceforth, the investigated nonlinear double boost converter system (16) is stabilizable by the controller (21) which has the parameter values calculated from (28). The proof is then completed. \(\square\)

Remark 0.2

It could be highlighted that the control parameters \(K_\alpha\), \(G_\alpha\), \(K_c\) and \(G_c\) are the key factors and Theorem 0.1 yields a sufficient criteria through LMIs, for guaranteeing the asymptotic stability of the above said augmented system (23). The appropriate selection of the positive constants \(v_1\), \(v_2\), \(v_3\), \(v_4\), \(v_5\), \(v_6\), \(\epsilon _1\), and \(\epsilon _2\) primarily influences the tracking accuracy and disturbance mitigation. To be specific, the stability of the above said augmented system (23) relies on the control key parameters \(K_\alpha\) and \(G_\alpha\), which are computed by tuning the values of \(v_1\), \(v_3\) and \(v_4\). Also, tuning parameters \(v_2\) and \(v_6\) enhances tracking accuracy and provides the suitable control parameters \(K_c\) and \(G_c\), whereas \(v_5\) enhances disturbance mitigation performance. Additionally, these tuning key factors supports to obtain a feasible solution of the derived LMIs (24)-(27). The appropriate values of \(v_1\), \(v_2\), \(v_3\), \(v_4\), \(v_5\), \(v_6\), \(\epsilon _1\), and \(\epsilon _2\) could be chosen by reducing the performance index known as Root Mean Square (RMS) \(J_e=\sqrt{\sum \limits _{\flat =1}^g \int \limits _{(\flat -1)P_e}^{\flat P_e}e^2(t) \textrm{d}t}\) (g is the number of periods). If the matrix inequality conditions (24)-(27) are not able to find feasible solutions, the adjustment in tuning constants yields the feasibility of those constraints, accordingly. Due to the fact that the suitable selection of constant parameters provides the feasibility of the matrix inequality conditions (24)-(27) and also reduces the tracking error e(t) tends to zero in steady-state, which shows the relationship between the constant parameters and the tracking performance.

Discussion

This section emphasizes the relevance of the recommended ESP based MRC technique by investigating the findings of simulation assessment for the delayed nonlinear double boost converter system undertaken. Particularly, the simulation results validate the derived theoretical results and also prove the supremacy of the recommended ESP based MRC technique when doing the comparative study with the existing methods such as HEID, IEID and PEID methods.

Now, the nonlinear time-delay system (16) influenced by delay and exterior disturbances is presented by using the mathematical model of double boost converter circuit model. In particular, the time lag and disturbances effect are existed in the control unit. Then, the system matrices for the considered nonlinear double boost converter system is taken as follows;

\(A=\begin{bmatrix} -160.5 & 0 & -7128.6 \\ 0 & -160.5 & -7128.6 \\ 399.2 & 399.2 & -199.6 \end{bmatrix}\), \(A_h=\begin{bmatrix} 0 & 0 & 0 \\ 0 & 0.1 & 0.02 \\ 0 & -0.03 & -0.1 \end{bmatrix}\), \(B=B_w=\begin{bmatrix} 17850 \\ 17850 \\ 0 \end{bmatrix}\), and \(C=\begin{bmatrix} 3.99&3.99&598.8 \end{bmatrix}\). The nonlinear term is \(N_u(t,x(t))=0.2\sin (x_1(t))\). The input delay is selected with \(\tau _u=0.2s\) and the non-differentiable state varying delay is chosen with \(h(t)=0.0183|\sin (t)|+0.1317\), here \(h_m=0.1317s\) and \(h_M(t)=0.15s\).

Case 1: Matched disturbance (input voltage variation)

To conduct the simulation assessment, select the varying disturbance voltage as \(w(t)=3|\sin (0.6\pi t)+\sin (0.4\pi t)+\cos (0.2\pi t)+\sin (0.5\pi t)|\) for \(t\le 0\), and the reference voltage is taken as \(r_{y}(t)= 100 [V]\). The MRC is designed with \(P_e=1s\) and \(\omega _c=100H\). Next, the coefficient sub matrices of the nominal converter model (18) \(A_{\alpha }\), \(A_{h\alpha }\), \(B_{\alpha }\) and \(C_{\alpha }\) has the same values as in nonlinear time-delay system (16) matrices, A, \(A_{h}\), B and C, individually. The filter \(F_\beta (s)\) is built with \(A_{\beta }=-0.01\), \(B_{\beta }=1\) and \(C_{\beta }=0.01\) whereas the enhanced disturbance estimator \(D_\gamma (s)\) is designed with the values \(A_{\gamma }=-0.5\), \(B_{\gamma }=1,\) \(C_{\gamma }=0.5,\) \(D_{\gamma }= 1\). The initial conditions are \(x_0=x_{\alpha 0}=\begin{bmatrix} 0.03&0.1 \end{bmatrix}^T.\) The constant parameters are taken possibly with the subsequent values: \(\nu _1=1\), \(\nu _2=1\), \(\nu _3=10\), \(\nu _4=0.000001\), \(\nu _5=0.000001\), \(\nu _6=1\), \(\epsilon _1=0.01\) and \(\epsilon _2=0.0001\). The saturation limit is taken as 100. Then, by finding the feasibility of the derived LMIs (24)-(27), the controller values are calculated from (28) as \(K_\alpha =\begin{bmatrix} 3.9162&3.9162&587.7369 \end{bmatrix}\), \(K_c=0.9906\), \(G_\alpha =0\) and \(G_c=152.5937\).

Fig. 8
figure 8

Tracking results with ESP and without ESP block.

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Fig. 9
figure 9

(a) Disturbance estimation and (b) Control input voltage.

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Fig. 10
figure 10

Tracking results with ESP and without ESP block subject to Backlash.

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Fig. 11
figure 11

Disturbance estimation and control signals subject to Backlash.

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Fig. 12
figure 12

Tracking results with ESP and without ESP block subject to dead-zone.

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Fig. 13
figure 13

Disturbance estimation and control signals subject to dead-zone.

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The results observed through the simulation study are demonstrated through the Figures 8,9,10,11,12,13. Particularly, Figure 8 displays the tracking performance with connected \(D_\gamma (s)\) in ESP and disconnected \(D_\gamma (s)\) in ESP, from which it could be noted that the ESP based MRC technique yields better tracking results when comparing with unplugged \(D_\gamma (s)\) in ESP block. Also, the disturbance estimation is shown in Figure 9(a) whereas the associated control input voltage is provided in Figure 9(b). To be specific, there are two control signals \(u_\alpha (t-\tau _u)\) and \(u(t-\tau _u)\) in Figure 9(b), which represent the control input without disturbance removal and with disturbance removal, respectively. However, the backlash non-linearity in control unit represents to the circuit components, specifically in gear changes, which results in time lag or circuit transmission loss. This backlash effect provides a nonlinear relationship among the control unit and the output of the circuit system, which complicates the control action. It should be needed to maintain the required control performance and the system stability which may be effected by backlash non linearity. The proposed ESP based MRC technique effectively handle the backlash effect in the system output. For the simulation purpose the above said disturbance w(t) is taken for \(10\le t\le 30\), and \(\text{ Backlash }(\varrho (u(t-\tau _u)))= -1.5\le u(t-\tau _u))\le 1.5\) are considered. The results from simulation assessment have been shown in Figures 10-11. In particular, it is observed from Figure 10(a) that the effects of backlash non linearity is reduced completely by using the ESP based approach and the backlash effect could seen in the system output as shown in Figure 10(b) when ESP is not plugged in the control input. In particular, from the Figure 10(b), it is possible to see that the output of the system deviates in two segments from \(t=0s\) to \(t=10s\) and from \(t=30s\) to \(t=40s\) where the disturbance input voltage is considered as zero. The associated disturbance estimation is also provided in Figure 11(a), which has only the slight deviation when comparing with the Figure 9(a). In addition, the corresponding control input signals given by \(u_\alpha (t-\tau _u)\) and \(u(t-\tau _u)\) indicating with and without disturbance elimination are plotted in Figure 11(b). In contrast, the dead-zone is an another type of non linearity which effects in control unit where changes in the control do not influence the system output. It may be possible in different types of actuators or controllers, sensors. Particularly, this could be intentional, used for preventing excessive sensitivity or unnecessary oscillations, or it might be an unwanted characteristic resulted by electrical circuit imperfections. The dead-zone in the saturation control unit is assumed as

$$\begin{aligned} \text{ Dead } \text{ Zone }(\varrho (u(t-\tau _u)))=\left\{ \begin{array}{cc} 0, & -1.5\le \varrho (u(t-\tau _u)) \le 1.5,\\ \varrho (u(t-\tau _u))+1.5, & \varrho (u(t-\tau _u))< -1.5,\\ \varrho (u(t-\tau _u))-1.5, & \varrho (u(t-\tau _u))> 1.5 \end{array} \right. \end{aligned}$$

By conducting the simulation assessment the related results are plotted in the Figures 12-13. To be specific, it is noted from Figure 12(a) that the dead-zone non linearity effect is completely diminished with the aid of recommended ESP and MRC based technique. Also, it could be seen from Figure 12(b) that the dead-zone effect occurs in the system output while the ESP block is not connected in the control input. The disturbance estimation is also given in Figure 13(a), which shows that there is only a small variation between disturbance and its estimation, when comparing with the Figure 9(a). Along with the dead-zone effect, it could seen that \(\varrho\) indicates the saturation input with the limits \(-100< \varrho < 100\). Also, the control signals such as \(u_\alpha (t-\tau _u)\) and \(u(t-\tau _u)\) representing with and without disturbance removal are shown in Figure 13(b). Thus, it is possible to have a conclusion from the above displayed simulation analysis that the recommended ESP based MRC technique yields accurate tracking performance and better disturbance reduction even there exist the non linear effects such as Backlash, dead-zone and saturation effect.

Case 2: Mismatched disturbance (grid voltage variation, random switching noise) without input delay

In order to demonstrate the excellence of the proposed ESP based MRC technique, the comparison analysis have been conducted with the state of art methods such as HEID, IEID and PEID discussed in45,46,47, separately. For the comparative study it is assumed that the disturbance input voltage exists due to some other factors in the double boost converter model undertaken. Thus, the disturbance voltage is mismatched and the related matrix can has the value \(B_w=\begin{bmatrix} 1000&0&0 \end{bmatrix}^T\) in the upcoming discussions. Also, the disturbance is considered as periodic square wave and \(r_y(t)=100 [V]\). By considering the above mentioned constant values and assuming the state delay as \(h(t)=0\), the LMIs (24)-(27) are solvable with the controller values from (28) as \(K_\alpha =\begin{bmatrix} 0.1095&0.1095&16.4395 \end{bmatrix}\), \(K_c=0.9240\), \(G_\alpha =0\) and \(G_c=338.5603\). The existing methods HEID, IEID and PEID discussed in45,46,47used internal model principle for tracking performance. As investigated in45,46,47, for the internal model parameters \(A_R=-0.01, B_R=1\) and EID low pass filter values \(A_F=-101,B_F=100, C_F=1\) the state feedback gain values are calculated using linear-quadratic-regulator (LQR) method. Then, for the LQR parameters \(Q_K=\text{ diag }\{1,1,1,1000\}\), \(R_K=1\), the computed feedback gain values are as \(K_P=\begin{bmatrix} -0.8217&-0.8217&-7.8469 \end{bmatrix}\) and \(K_R=31.6226\).

Fig. 14
figure 14

Tracking performance for the grid voltage variation based on the subsequent approaches: (a) ESP (b) HEID (c) IEID (d) PEID.

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Fig. 15
figure 15

Disturbance estimation for the grid voltage variation based on the subsequent approaches: (a) ESP (b) HEID (c) IEID (d) PEID.

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Fig. 16
figure 16

Tracking performance for the random wave disturbance voltage based on the subsequent approaches: (a) ESP (b) HEID (c) IEID (d) PEID.

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Fig. 17
figure 17

Disturbance estimation for the random wave disturbance voltage based on the subsequent approaches: (a) ESP (b) HEID (c) IEID (d) PEID.

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Next, the observer gain Lbased on HEID method is calculated by solving the LMI (38) in Theorem 1 of45 as \(L=\begin{bmatrix} 19873&19873&0 \end{bmatrix}^T\). The observer gain based on IEID method is obtained by solving the dual system in46 with the aid of LQR method by taking the values \(Q_L=\text{ diag }\{1,1,1000\}\) and \(R_L=1\). The resulted observer gain matrix is \(L=\begin{bmatrix} 1.0846&1.0846&30.6834 \end{bmatrix}^T\). Finally, the observer gain based on PEID approach is calculated by using LQR method as investigated in47 and \(L=\begin{bmatrix} -0.0069&-0.0069&31.3053 \end{bmatrix}^T\). By considering these controller designs based on HEID, IEID and PEID methods, the comparative simulation study is investigated and the observed results are displayed in Figures 14,15,16,17. Specifically, the tracking performance based on ESP, HEID, IEID and PEID methods are provided for the periodic square wave disturbance (which indicates the grid voltage variation) in Figures 14(a),14(b),14(c),14(d), respectively form which it is noted that ESP based MRC technique yields better tracking performance while comparing with HEID, IEID and PEID methods. The corresponding mismatched disturbance estimations regarding ESP, HEID, IEID and PEID methods are presented for the periodic square wave disturbance input voltage in Figures 15(a),15(b),15(c),15(d), accordingly which portrays that ESP based MRC technique outperforms the HEID, IEID and PEID based disturbance elimination methods. Similarly, the tracking performance analysis based on ESP, HEID, IEID and PEID techniques are presented for the random square wave (which indicates the switching noise) in Figures 16(a),16(b),16(c),16(d), separately. These Figures demonstrate that the internal model based HEID, IEID and PEID methods provide poor tracking performance in the case of mismatched random square wave disturbance input voltage when comparing with our proposed ESP based MRC technique. The related mismatched disturbance estimations using ESP, HEID, IEID and PEID methods are displayed in Figures 17(a),17(b),17(c),17(d), accordingly which states that ESP based MRC technique provides dominance performance in disturbance elimination comparing with the existing HEID, IEID and PEID based disturbance reduction approaches.

Case 3: Mismatched disturbance (due to load variation) with input delay

Next, one more scenario is considered, in which load variation disturbances caused by either abrupt or unforeseen variations in the electrical demand affects the PV system performance, particularly the battery and load. These disruptions can rise from different dynamic concerns during process and could negatively affect the stability, competence, and PV system control performance. In order to describe the efficiency of the proposed ESP based MRC technique against the input delay and load variation disturbances, the comparison analysis is studied with the existing EID based SP approach23. For conducting the comparison simulation analysis, the LMI (24) of Theorem 3.1 in EID based SP method23 could be solved with the double boost converter system matrices and the observer gain is computed as \(L=\begin{bmatrix} -0.0518&-0.0518&-0.1765 \end{bmatrix}^T\).

Figure 18
figure 18

(a) Tracking results based on ESP-MRC and EID-SP methods (b) Disturbance due to load variation and its estimation based on ESP-MRC and EID-SP methods.

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Table 1 Performance index for comparative study based on ESP-MRC and EID-SP methods.
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The related simulation results are provided in Figures 18(a),18(b), separately. It is obviously noted that the tracking results and the disturbance estimation results for the mismatched disturbance due to load variation could be achieved perfectly with the implementation of the proposed ESP based MRC technique when comparing with the EID based SP method. The performance index for comparative study is provided in Table 1, which indicates that the performance index for tracking error and disturbance estimation error based on Integral Absolute Error (IAE), RMS, peak-to-peak value (PPV) regarding the proposed ESP based MRC method is smaller than the existing EID based SP technique.

Case study analysis

Based on above discussions, the comparison analysis together with existing controller methods such as EID, IEID, PEID and conventional SP describes the dominance of proposed ESP based MRC design in the case of accurate tracking and disturbance attenuation performance for double boost converter system. In particular, EID based SP method provides better disturbance mitigation only for matched disturbance such as input voltage variation whereas the proposed ESP based MRC technique yields better disturbance mitigation for both matched disturbance like input voltage variation and mismatched disturbances like grid voltage variation, switching noise represented by random square wave and load variations indicated by trapezoidal wave, accordingly. Thereby, it could be noted from the above investigated simulation findings that the suggested ESP based MRC technique provides tracking accuracy as well as better disturbance attenuation, though there exist the mismatched disturbances scenario.

Conclusion

In this work, the ESP based MRC technique is proposed to achieve the accurate tracking performance of delayed double boost converter with saturation nonlinearity, disturbances and input delay. By considering the combination of LKF and LMI methods, a group of matrix inequality conditions are derived for ensuring the asymptotic tracking performance of the delayed double boost converter under study. The derived feasible solutions of the LMIs provide the design of controller parameters. The simulation analysis and the related outcomes obtained from a double boost converter model obviously validates the practicability and efficiency of the suggested control system. The obtained results conclude that the recommended control technique not only ensures asymptotic stability but also significantly improves tracking accuracy compared to conventional HEID, IEID and PEID methods. The resulted control system setup effectively regulates the power flow, guarantees precise alignment with the provided output and reduces oscillations which demonstrate its effectiveness in both steady-state and dynamic conditions. Thus, the proposed control framework is simple and effective because of its accurate tracking capacity, as provided by the simulation outcomes using a double boost converter model. Also, the comparison analysis with the exiting HEID, IEID, PEID and EID based SP methods , the proposed ESP based MRC technique yields better tracking performance and disturbance elimination.