Robustness Enhancement of Feedback Control in DC/DC Converters

Version	Summary	Created by	Modification	Content Size	Created at	Operation
1		Asimenia Korompili	--	6405	2023-06-24 21:09:00	\|
2	format	Jason Zhu	-4 word(s)	6401	2023-06-25 03:59:35	\|

This entry is adapted from the peer-reviewed paper 10.3390/en16124563

One major limitation that feedback control methods exhibit, to different extent, is the low robustness against disturbances, which deteriorates their performance. For the enhancement of the robustness, various solutions have been proposed in literature, such as adaptive control structures or the integration of disturbance observers. These approaches to enhance the robustness of the feedback controllers are discussed in a systematic way. Researchers focus on the technologies that offer robustness in a generic baseline feedback controller, rather than on the modification of a specific feedback control method. Three extensively utilised approaches are reviewed, by presenting their principles and common structures in their application for the voltage regulation in DC/DC converters.

DC/DC converter feedback control

1. Adaptive Control

The main goal of the adaptive control is to adjust the behaviour of the baseline feedback controller in response to disturbances, to retain the nominal performance. To achieve this, the adaptive mechanism processes, in real time, input/output measurements of the real converter and designs/adjusts online the feedback controller or chooses the appropriate control from a set of controllers ^[1]^[2]^[3]^[4]. In this way, the adaptive control structures present inherent robustness against disturbances occurring in the real converter, while maintaining acceptable stability margins without performance degradation. For example, this is demonstrated through frequency-domain analysis in the case of a buck converter with an adaptive loop added in the conventional voltage control loop in ^[5]. This is opposite to the baseline feedback controllers presented above, which are fixed-gain, static control structures, causing degradation of the performance when applied in the real converter. In other words, the baseline feedback controllers heavily rely on the nominal converter model used for the design, which is an ideal model without disturbances, whereas the adaptive control structures are model-independent. Over the last decades, the field of adaptive control has experienced tremendous progress in theory. Ref. ^[6] traced the developments from a historical perspective by presenting the problem statements and key solutions in the field.

The online adaptation mechanism is designed according to the class of the adaptive control scheme. In the most widely applied class of the identifier-based adaptive control, this mechanism includes an online parameter estimator, also referred to as parameter identifier or adaptive law, and the online adjustment of the feedback controller according to the estimated control parameters. In the class of the nonidentifier-based adaptive control, the adaptive mechanism consists of stored feedback control models and an appropriate logic for selecting the right one in real time ^[2]. In many studies, gain scheduling is also considered as an adaptive control scheme, where the adaptation mechanism includes just a look-up table with a schedule logic ^[2].

The identifier-based schemes can be designed in two different approaches, according to the way the online parameter estimator is combined with the feedback control law: the indirect or explicit adaptive control and the direct or implicit adaptive control ^[2]. The former is more difficult in application, as the solution of the algebraic equations for the online control design is not guaranteed at each time (stabilisability problem) ^[2], which can be critical for the operation of the converter. The latter is suitable only for converters of the MP system class ^[1], and thus not applicable for the direct voltage control of DC/DC converters in the boost stage. Combined direct/indirect adaptive control schemes have been proposed, which solve the stabilisability problem of the indirect approach and relax the MP assumption of the direct approach. However, this is achieved at the cost of higher complexity of the resulted adaptive control structure ^[1], which makes more difficult its implementation in the converter hardware.

The most popular identifier-based schemes are the model reference adaptive control (MRAC) and the adaptive pole placement control (APPC) ^[1]^[2]. For example, an MRAC scheme is combined with an LQG controller in ^[7], to adjust its control parameter, for ensuring the stability and robustness against converter model uncertainties, as discussed in Section 2.6 of the associated paper. The main limitation of the MRAC is its nonapplicability to converters of the NMP system class. On the other hand, APPC is suitable for both MP and NMP systems ^[2]. This scheme is usually referred to as self-regulator or autotuning scheme. Ref. ^[8] classifies the auto-tuning scheme as an adaptive control that is enabled and executed at discrete intervals upon event detection, regularly scheduled interval or external command. However, the term of auto-tuning is used more widely also for adaptive mechanisms that update the parameters of the feedback controller continuously, to accommodate changing operating point or external disturbances. The adaptation mechanism consists of methods, such as Ziegler–Nichols, fuzzy logic, or genetic algorithms, that evaluate, in real-time the control performance according to predefined indicators and metrics, such as damping factor and natural frequency of oscillations, to adjust the feedback controller parameters for achieving the desired behaviour determined by the control objectives ^[3]. Examples of the application of such APPC schemes in the DC/DC converters include the introduction of autotuning adaptation mechanisms in the structure of baseline feedback controllers, to adjust their parameters in response to real-time measurements; selectively, some are listed here ^[9]^[10]^[11]^[12].

For the online parameter estimation, system identification (SI) techniques have been integrated in the identifier-based adaptive control structures. Ref. ^[13] reviewed SI techniques suitable for DC/DC converter applications. Nonparametric SI techniques, such as correlation estimation ^[14] and power spectrum density methods ^[15], have the advantage of no need for prior knowledge of the converter model. However, these techniques usually present low accuracy and speed of the parameter estimation, high sensitivity to disturbances, and high computational complexity ^[13]. Therefore, the feasibility of their application should be investigated at each case. Further research is required for reducing the computational complexity and the impact of disturbances on the accuracy of the parameter estimation. For example, Ref. ^[14] proposed an analog cross-correlation-based parameter computation, which has lower computational complexity and data storage requirements in comparison to digital correlation implementations. Parametric SI techniques have gained application interest, although these require a definition of the converter model in advance ^[16]. The selected converter model is always application-dependent and the complexity of the SI technique is subject to the modelling approximations ^[16]. Iterative and recursive estimation algorithms, such as least mean squares ^[17], recursive least squares ^[16] and Kalman filter ^[18], have been integrated in the adaptive control of DC/DC converters, providing simple adaptive mechanisms with higher convergence speed, higher accuracy of parameter estimation, and lower sensitivity to disturbances than the nonparametric SI techniques ^[13]. The main issue of these parameter estimation algorithms, especially the recursive ones, is the high computational burden due to the large number of needed mathematical operations, which may require a high-specification microprocessor for the successful implementation in the converter hardware. Hence, these techniques have not been fully exploited and adopted for the online parameter estimation in adaptive control in low-cost, low-power converter applications ^[16]. There are research attempts to reduce the computational complexity of these techniques, such as through the integration of dichotomous coordinate descent, which offers a fast, computationally light adaptation mechanism ^[16]. Ref. ^[17] proposed a state-space-based parametric SI technique that reduces the computational burden in comparison to transfer function-based techniques, since fewer parameters have to be estimated. Ref. ^[18] proposed partial update methods applied to the Kalman-filter-based SI to reduce the computational burden of the identification algorithm. Noniterative estimation methods, such as limit cycle oscillation ^[19] or relay schemes, present low computational burden, but suffer from low parameter estimation accuracy and cause oscillations of large amplitude in the output voltage of the converter during the identification process ^[13]^[17]. The parametric SI techniques can be easily applied in adaptive control structures such as APPC and MRAC for the online parameter estimation, and can be implemented directly in digital control designs, eliminating the errors of domain transformation ^[13]. However, many factors regarding the efficient implementation of the iterative estimation algorithms, like their step size, are still not discussed in literature ^[13]. The introduction of new-generation microprocessors allows the investigation of the capabilities of alternative SI techniques. In general, further research in the area of online parameter estimation through SI techniques is required to further reduce the computational complexity and the digital hardware usage and allow compact low-cost implementations while achieving parameter estimation of high speed and accuracy. This is imperative for the wider adoption of adaptive control in the field of DC/DC converters.

Although the adaptive control approach is adopted to enhance the robustness of the baseline feedback control, the parameter estimation of the adaptive mechanism is also sensitive to converter model uncertainties, as its design relies on the nominal converter model. This can drive the adaptive controller unstable. Modifications of the online parameter estimation have been proposed in order to enhance its robustness, such as

σ

-modification or leakage method and the use of a dead zone in the adaptation ^[1]. However, even these robust adaptive control methods present drawbacks, such as slow parameter adaptation leading to slow transient response of the converter controller and chattering due to dead zone of the adaptive mechanism ^[4]. The improvement of the performance of robust adaptive control is still an open research problem, as the proposed modifications in literature are application-dependent and rely on the ideal performance of other elements of the control structure ^[4].

The stabilisability problem of the indirect adaptive control and the robustness issues of the online parameter estimation do not occur in the class of the nonidentifier-based adaptive control ^[2]. The major advantage of this switching-based adaptive control is the rapid adaptation to sudden disturbances, which can occur during the converter operation ^[3]. One approach of this class is the multiple-model-adaptive control (MMAC) scheme, where a switching approach, based on a supervisory control logic, selects a feedback controller from a finite set of controllers based on the input/output measurements; the set of controllers is designed according to a set of converter models. Robust MMAC (RMMAC) schemes have been developed, where the feedback controller set is designed with robust control techniques, to account for robust stability and performance requirements ^[2]. Despite the good performance of the scheme, there are drawbacks related to the assumptions for the design of the models set (disturbance model, especially stochastic one, initial conditions, etc.), as well as stability limitations ^[2]^[3]. Another approach of nonidentifier-based adaptive control is the unfalsified adaptive switching control (UASC), which relies solely on input/output data to choose the feedback controller from a given set of controllers, without requiring any converter model. This provides a model-free adaptive control, thus relaxing thus the assumptions of the converter modelling included in MMAC ^[2]^[3]. The main drawback of this adaptive control scheme is the lack of guarantee that the suitable feedback controller is chosen. The multimodel UASC (MUASC) scheme, where nominal converter models can be pairwise associated with candidate feedback controllers, can reduce the switching between different controllers and the chance that destabilising controllers will be selected. This can reduce the transients in the converter performance, which cannot be guaranteed in schemes without nominal converter models ^[2]. More advanced adaptive control schemes mix the identifier- and nonidentifier-based approaches to take advantage of positive features of both classes. Adaptive mixing control (AMC) can mix/combine the outputs of candidate controllers, providing the advantage of smooth transition (interpolation) from one controller or combination of controllers to another ^[2]. This adaptive control scheme achieves accurate parameter estimation and thus stability of the closed-loop system. However, the interpolation of the control outputs, instead of discontinuous switching logic between control candidates, might render this adaptive control scheme slower to sudden disturbances than the classical switching-based adaptive control (nonidentifier-based adaptive control class).

In general, (robust) adaptive control schemes can effectively deal with large-signal disturbances, strongly enhancing the robustness of the baseline feedback controller of the structure. One main limitation is that all schemes rely, to different extent, on online learning of the real converter plant, to design or choose the suitable controller. Since the adaptive control is a highly data-driven approach, it relies on the quality of input/output data to be processed for this online learning ^[4]. Corrupted data by disturbances can lead to the design or selection of a feedback controller that cannot stabilise the DC/DC converter, causing large transients in the converter operation ^[2]. Different adaptive control schemes present different sensitivity to corrupted data and thus different probability to result in transients in the different applications. Further understanding of the difficulties associated with the process of controlling a converter, while trying to simultaneously to learn the parameters of its mathematical model from input/output data, is essential for the exploration of possible combination between adaptive mechanisms and baseline feedback control methods ^[2]. Furthermore, more complex approaches at the intersection of adaptive control and artificial intelligence techniques, such as machine learning, reinforcement learning and neural networks, have been proposed recently, leveraging on the suitability of those techniques in approximating the nonlinear functions of the real converter plant, without the need for “learning” a complex mathematical nominal model of the converter dynamics ^[3]^[6]. For example, in ^[20] a deep reinforcement learning technique is adopted for the online adjustment of the parameters of the sliding mode observer-based PI control of a buck/boost converter. The computational burden and training time and data are critical factors to be taken into account for the wide application of these promising solutions of model-free-adaptive control ^[4]. The advancements in microprocessors and high-speed computing technology will be proven crucial to enable further development and practical implementation of adaptive control schemes on high-bandwidth converter hardware ^[3]. Although active for several decades, the field of adaptive control is still a fertile research area with many immature aspects to be further investigated and clarified, such as convergence guarantees and methods for robustness analysis.

2. Robust Control

Robust control aims at designing feedback control laws that guarantee closed-loop stability and desired performance not only for the nominal converter model, but also for the real plant, with respect to a class of model uncertainties encountered in real life. The robust control design method incorporates the knowledge of the converter model uncertainties, as well as robustness and performance requirements, expressed as frequency-dependent weighting functions, into the feedback control design process. This provides a systematic approach to design the converter controller for high inherent robustness against model uncertainties ^[21]^[22]^[23]^[24]. The weighting functions allow to design of the controller so that the converter achieves the desired performance in a certain frequency range of its operation, whereas it presents robustness against uncertainties appearing in other frequency ranges, e.g., frequency of unmodelled dynamics ^[21]^[22]^[23]. This incorporation of frequency-domain uncertainty models and performance requirements in the design of robust control is related to the ∞-norm of the weighted closed-loop transfer function of the converter, as explained in Section 2.6 of the associared paper: the ∞-norm bound of the weighted closed-loop transfer function provides a sufficient condition for robust stability and performance. Precisely, the design methodology of robust control, called

μ

-synthesis, integrates the baseline feedback controller of the

H_{\infty}

optimisation problem for synthesis and the structured singular value

μ

for analysis. The robust control can deal with uncertainty models of different level of structure. The

μ

-analysis is able to handle structured and unstructured model uncertainties, opposite to the

H_{\infty}

optimal control, which is not able to handle robustness bounds and performance measures associated with structured uncertainties.

Robust control manages to optimise the robustness and performance characteristics of the baseline feedback control, thus achieving thus robust stability and performance. This benefit is demonstrated in several applications in DC/DC converters. Ref. ^[25] provides a detailed description of uncertainties in DC/DC converters and presents the theory and procedure of the

μ

-synthesis of the robust control for a boost converter. In the same application, Ref. ^[26] compares the performance of the robust controller with this of the PI controller in frequency domain, demonstrating the superiority of the former in transient response and disturbance rejection. Similar comparison conclusions are also drawn in ^[27] for the case of a buck/boost converter. Ref. ^[28] demonstrates the superiority of the

H_{\infty}

voltage control, assisted by a constant-gain current control loop, over the classical peak current mode controller in the case of a boost converter. The robust control exhibits better transient response under disturbances, although the order of the resulted feedback controller from the iterative technique is reduced, which is necessary to facilitate its practical implementation in the converter hardware. Full and reduced order

H_{\infty}

robust control was adopted for the design of PI control for the boost converters in a hybrid power generation system in ^[29]. Time- and frequency-domain analyses demonstrate the higher robustness in comparison with the classical PI control.

The key point of the robust control is the uncertainty model and its validation. Further research is required to gain insight into the influence of the accuracy of the uncertainty models on the achieved robustness properties of the converter controller ^[21]^[23]. A systematic approach to expand the robust control design framework to include system identification, which can provide both nominal converter and uncertainty models, can be a major step forward for its wide application ^[23]. Moreover, the developed techniques for the robust control deal with the complex

μ

problem that considers complex values of uncertainties, which is a reasonable assumption only for uncertainties representing unmodelled dynamics. However, these techniques cannot deal well with real values of uncertainties, such as parametric uncertainties, resulting in conservative robust control designs. To handle the mixed

μ

problem, where the uncertainties can have both real and complex values, Refs. ^[21]^[30] suggest the use of alternative iterative techniques for the design of the controller K. Although the mixed

μ

problem is more realistic approach in the case of converters, where the model uncertainties can appear as real parametric uncertainties, while the performance specifications are modelled as fictitious complex weighting functions, the developed techniques are not widely applied. More work is necessary to provide insight in their features, advantages, and limitations in their applications in the field of DC/DC converters.

3. Disturbance and Uncertainty Estimation and Attenuation (DUEA)

DUEA is a family of control techniques sharing the same fundamental idea of dealing with disturbances in the real converter: in addition to the baseline feedback control action, there is a mechanism that estimates the total disturbance and rejects this directly through a feedforward cancellation action ^[31]^[32]^[33]^[34]. The total disturbance is a lumped disturbance that refers to both external and internal disturbances. Although these control techniques became known with the name DUEA, the aim is not only to attenuate/suppress the disturbance, but to cancel it directly in the sense of totality and finality, i.e., after the disturbance rejection, there is no effect of it on the operation of the converter, thus achieving the absolute invariance. In that sense, this principle of disturbance rejection is similar to the invariance principle. For this reason, the DUEA techniques are also referred to as disturbance rejection control (DRC) techniques ^[32]. They are also mentioned as active antidisturbance control. This is opposite to the passive antidisturbance control approach of feedback control structures without the feedforward disturbance cancellation. The passive antidisturbance control in DC/DC converters is driven only by the difference between the output voltage and its set-point, with the goal to attenuate the disturbance that tends to drive the output away from this set-point. Hence, it cannot react directly to disturbances and thus it can only compensate for them in a relatively slow manner, opposite to the DUEA approach ^[31]^[32].

The generic structure of DUEA (or DRC) consists of the disturbance rejector and the feedback controller. The disturbance rejector includes the state and disturbance estimation and the disturbance cancellation ^[32]. The composite control action (control input) consists of two terms, a feedback term generated by the feedback controller and a feedforward term generated by the disturbance rejector ^[31]^[33]. Hense, the control action is shared between the feedback controller and the disturbance rejector. The idea is that the rejector estimates and cancels, in real time, the total disturbance, to transform the real converter plant to the enforced plant, which is the nominal converter model. This enforced plant, which is a disturbance-free converter model, represents the dynamics to be controlled by the feedback controller ^[32]. The inner feedforward loop is thus designed for the control objective of disturbance rejection, and the feedback control loop is designed independently for the enforced plant, meeting the control objectives of nominal performance ^[31]^[33]. This two-degrees-of-freedom control structure allows the meeting simultaneously conflicting control objectives. The separate design of the two loops also provides design flexibility to the DUEA structure, as various disturbance estimation methods can be integrated in various baseline feedback controllers.

The notions of the total disturbance, the disturbance rejector loop, and the enforced plant bring benefits for the design of the DUEA structure. Since the model uncertainties are cancelled as part of the total disturbance, there is no need for a high-fidelity model of the real converter to design the feedback controller. This means that in DUEA structures, the complexity of the control design problem, associated with nonlinear, uncertain, time-varying converter models, can be decreased ^[32]. Moreover, the conventional boundaries of system classes (linear or nonlinear, time-varying or time-invariant, MP or NMP, etc.), setting apart different control methods, are completely dissolved ^[35]. The feedback controller can be designed according to a simple nominal converter model by simply applying any modern control method, without the concern of possible robustness issues, as the disturbance rejector cancels the model uncertainties. In addition, by assuming the natural couplings of the real converter as part of the total disturbance to be rejected, the DUEA provides decoupling control, forming the converter model as a simple input/output relation to be used for the feedback control design ^[36]. This approach is advantageous in terms of design time and cost, considering many existing decoupling control approaches, where the accurate mathematical model of the real converter plant is needed for formulating and controlling the cross-couplings ^[37]^[38]. For example, such a control approach provides effective decoupling control of power flows in a four-port multi-active bridge converter in ^[39], requiring information only about the system order of the converter.

In that sense, the DUEA techniques are model-independent control design approaches, needing only information about the system order of the converter, as in model-free control ^[32]^[36]^[40]. In practice, this means that the gains of the feedback controller are independent of the converter plant, as it is designed for the simple, disturbance-free nominal model. This allows the same feedback controller to be applied in different DC/DC converters. The limit of this general applicability is determined by the timescale of the converter that reflects its dynamics. Different converters with the same timescale can be controlled by the same controller of the DUEA control structure, without any gain tuning or other modification. Even more, the control gains can be adjusted easily between converters of different timescales ^[41]^[42]^[43]. Furthermore, exactly because the total disturbance is estimated and cancelled, there is no need for large value of the gain of the feedback controller for disturbance rejection, as in the concept of the high-gain control ^[44]. In addition, opposite to the control methods relying only on feedback loop, in the DUEA structure the feedback control does not need to include additional integrators to achieve accurate reference tracking, as the needed integral action is provided through the disturbance rejection.

These benefits for the DUEA design occur thanks to the disturbance rejector loop. The fast and accurate estimation of the total disturbance, including as much frequency information as possible, is critical for the good performance of the DUEA applied in DC/DC converters due to the fast dynamics in DC systems. On the other hand, the transfer of high-frequency noise from the sensors of the converter into the control signal, through the disturbance estimation, can easily cause actuator saturation and thus decrease of the control quality. Therefore, the proper design of the disturbance rejector, to compromise between speed and accuracy of the disturbance estimation and sensitivity to noise, is essential for the good performance of the DUEA when applied in DC/DC converters. Moreover, the better the disturbance estimation and cancellation in the rejector, the simpler the enforced nominal converter model can be considered for the feedback control design ^[32]. On the other hand, the more accurate the converter the model is considered in the design of the disturbance rejector, the lesser the computational burden and the latency of the disturbance estimation. The more accurate and fast the disturbance estimation, the more effective the disturbance rejection. This leads the performance of the feedback controller to be closer to the nominal, with fast transient response and accurate reference tracking. Considering these relations of converter modelling, disturbance estimation and rejection, and performance of the feedback controller, the quality of the DUEA control structure depends on the balance between the design of the disturbance rejector and the design of the feedback controller, according to the benefits and costs of these designs ^[32].

The disturbance rejection makes the DUEA techniques more anticipatory than the error-based PID control. Therefore, the DUEA techniques present significant reduction in the control effort. In PID control, this can be addressed through feedforward terms, which have to be customised for each converter and thus depend strongly on the knowledge about the converter and the disturbance ^[36].

Apart from the various feedback control methods that can be applied, the disturbance rejector can also take various forms. Considering the different disturbance rejectors, various DUEA techniques have been developed, such as disturbance observer-based control (DOBC), extended state observer (ESO) in active disturbance rejection control (ADRC), uncertainty and disturbance estimator (UDE)-based control, unknown input observer (UIO) in disturbance accommodation control (DAC), equivalent input disturbance (EID) estimator-based control, and generalised proportional-integral observer (GPIO)-based control ^[33]. Although these DUEA techniques have been developed starting from different perspectives and for different application purposes, they present strong conceptual similarities. Indeed, they can all be reduced to the basic structure of the DUEA (DRC), where the disturbance estimation and cancellation are now unified in the notion of the disturbance rejector ^[32]. Hence, the DUEA structure acts as an umbrella platform, a framework, which encompasses all aforementioned methods. Moreover, it provides a generalised principle, according to which various control solutions can be categorised and their similarities and differences can be organised, especially in terms of the nature of the total disturbance and how this is treated for its estimation and cancellation. Through the DUEA framework, previously scattered work in the field of disturbance rejection can now be sensibly understood and reconnected in a systematic way to reveal its full potential ^[36].

The most common DUEA techniques are the DOBC and the ADRC; the rest of the techniques present strong similarities with these two main DUEA structures ^[33]. The filter is the most critical element of the DOBC, which is designed considering the robustness objective, as well as bandwidth limitations of the feedback controller and the converter ^[45]^[46]. The filter design can also be extended for converters of the NMP system class as described in details in ^[47]^[48]^[49]. Guidelines for the filter design, in terms of bandwidth constraints of the disturbance rejector, are provided in ^[50]^[51] for MP and NMP systems. To exploit the (partially) known nonlinear dynamics of the converter during the design of the DOBC structure, improving the estimation and cancellation of the disturbance, and thus the performance and robustness of the controller, nonlinear DOBC (NDOBC) techniques have been developed ^[31]^[33]. Design guidelines for full-order and reduced-order NDO in NDOBC structures were proposed in ^[52]^[53]. Ref. ^[54] proposed NDOBC suitable for matched and mismatched disturbances, rejecting totally the disturbance and thus providing thus higher robustness of the closed-loop system, while achieving the nominal performance. Higher-order NDO have also been proposed to exploit more structure information of higher-order disturbances, and thus facilitate their estimation and cancellation ^[31]^[33]. Ref. ^[55] presented a historical review of the main developments of the DOBC models.

(N)DO and disturbance cancellation actions have been integrated into many baseline feedback controllers, forming (N)DOBC structures applied for voltage control in DC/DC converters. The (N)DO provide total disturbance rejection, and thus robustness enhancement, while keeping the nominal performance of the closed-loop system of the converter. In many of these works, like in ^[56]^[57], the formulation of the system with lumped disturbances in both dynamic equations of the converter or its coordinate transformation allows the estimation of both matched and mismatched disturbances affecting adverse the voltage regulation of the converter. This is independent from the type of the observer (e.g., finite-time, I&I, etc.). The estimated disturbances are then integrated directly in feedforward channels in the design of the feedback controller (e.g., backstepping control, SMC) to cancel their effect on the controller performance. In other cases, like in the DOBC structure of a buck converter in ^[58], a compensation gain is needed for the feedforward term of the mismatched disturbance in the final control law.

The generic structure of the ADRC consists of the ESO, the tracking differentiator (TD) and the feedback controller. The ESO estimates the physical states of the converter together with the virtual state of the total disturbance. The estimated total disturbance is rejected by the feedforward term of the control input, passing through the disturbance cancellation block. The feedback term of the control input is generated by the feedback controller, which regulates the estimated states of the converter to the state reference trajectory provided by the TD ^[33]^[44].

The ESO can be designed as linear or nonlinear, with the former presenting larger convergence speed due to shorter computation but the latter presenting higher estimation accuracy. Appropriate design of a time-varying gain of the ESO can also facilitate the estimation accuracy ^[44]. The accepted noise sensitivity should be also considered as restriction for the ESO design ^[59]. Ref. ^[60] enhanced the linear ESO (LESO) in a DC/DC converter with a correction function, which can estimate a wider spectrum of frequencies of the total disturbance. This offers a more accurate disturbance estimation for various forms of disturbances common in the field of converters. Ref. ^[61] introduced the differential signal of the total disturbance as a new state in the traditional LESO of a DC/DC converter, achieving the tracking of the dynamics of the total disturbance and thus the improvement of the estimation accuracy and speed. In a more advanced ADRC model of DC/DC converter, Ref. ^[62] proposed a cascaded ESO based on a virtual decomposition of the total disturbance, where each ESO level is responsible for estimating different frequency range of the disturbance. In this way, the accurate estimation of the total disturbance is achieved, while the measurement noise suppression is increased, opposite to the traditional single-level high-gain ESO. This ADRC model with the cascaded ESO structure belongs to the more general category of the composite hierarchical antidisturbance control, where multiple disturbance rejector loops are used for rejecting different types of disturbances such as stochastic noise and deterministic disturbances due to unknown parameters of the converter filter. The guaranteed stability of such control structure is the main advantage, with the high complexity due to the coupling of the disturbance rejector loops being the major concern ^[31]^[33].

The TD offers a transient profile that the converter states should track. This reference profile changes gradually and thus smoother, without step jumps. This allows for a more aggressive control design, for example, with higher gains, while avoiding actuator saturation ^[44]^[63]. In addition, since the state reference trajectory is obtained through integration, it is less sensitive to noises ^[63]. Nonlinear TD can be applied for the generation of more accurate state reference trajectory; however, linear TD performs better in the presence of measurement noise ^[44]. Ref. ^[64] presented the design conditions of the three components of the nonlinear ADRC that guarantee stability, disturbance rejection, and reference tracking.

The original ADRC formulation, where the ESO is expressed in integral-chain form (canonical form) and the disturbance cancellation is a direct feedforward term, without any particular gain, is applicable only to converters of the MP system class and can reject only matched disturbances ^[65]^[66]. A generalised ADRC model has been proposed, where the ESO incorporates knowledge about the converter plant, and the disturbance cancellation action includes a compensation gain. The former makes the generalised ADRC applicable also to the NMP systems, whereas the latter makes it suitable for also rejecting mismatched disturbances ^[65]^[66]^[67]. Hence, the generalised ADRC is applicable to a wider class of converter plants and disturbance types than the original ADRC, presenting advantages in terms of disturbance rejection and performance of the closed-loop system of the converter ^[68].

The more general form of the nonlinear ADRC (NL-ADRC) exhibits, in most of the cases higher robustness to disturbances and better dynamic performance than the linear ADRC (L-ADRC). However, the tuning methods for NL-ADRC are limited in practical applications, because they require significant design effort and time and usually ignore physical limitations of the actuator, such as bandwidth and noise. Moreover, the theoretical analysis is a difficult task in NL-ADRC ^[69]. Ref. ^[70] used the describing function method to approximate the NL-ADRC with a linearised model, through the concept of the equivalent gain, to analyse the stability, transient response, and reference tracking in frequency domain. The L-ADRC is more transparent to practicing engineers, and thus much more applied in DC/DC converters. For example, Ref. ^[71] applies a L-ADRC in a boost converter and relates it to the precursor PI control, by comparing the two approaches through analysis of the achieved disturbance rejection in frequency domain. Ref. ^[72] integrates a reduced-order ESO to the proportional controller of a buck converter and analyses the robustness of the control structure against model uncertainties in frequency domain. However, there are still only few theoretical works for the analysis of the L-ADRC performance in the frequency domain with which practising engineers are more familiar ^[73]^[74]^[75]. Ref. ^[69] compared L-ADRC and NL-ADRC in terms of tuning ease, as well as stability and performance analysis. To integrate the merits of the L-ADRC and NL-ADRC, it also proposed a switching control scheme between L-ADRC and NL-ADRC and analysed its stability.

For promoting the application of NL-ADRC models in DC/DC converters, research activities focus on the online tuning of the parameters. For this purpose, heuristics algorithms, such as particle swarm or ant colony optimisation or fuzzy control, have been applied in the ADRC structure of DC/DC converters for the online adaptation of the control parameters, like in the case of the NL-ADRC structure of a bidirectional DC/DC converter in ^[76]. Further research for the online tuning of the ADRC parameters in DC/DC converters can be inspired by such applications in the similar systems of induction motors and hybrid active power filters ^[77]^[78]^[79]. Up until now, the NL-ADRC models applied in DC/DC converters usually refer to the abstract mathematical formulation, whose performance depends only on the tuning of the parameters. Although the ADRC approach allows and even promotes the incorporation of well-analysed existing methods for the estimation and control, as the general DUEA framework, there are only few works in which modern methods of observer or feedback controller are integrated in the ADRC structure when applied in DC/DC converters. For example, in ^[80] the NL-ADRC structure of a boost converter consists of a backstepping feedback controller and a generalised proportional-integral observer (GPIO). A reduced-order GPIO is also used in the ADRC model of a buck converter in ^[81], to enable easier practical implementation comparing to the traditional full-order ESO. In that work, the feedback controller consists of an optimal control with output voltage prediction, which enables the inclusion of performance indexes in the design of the ADRC voltage controller of the converter.

Although the ADRC is just a particular rendition of the DUEA methods, in the literature it is considered their spearhead, and in many works the concept of the DUEA (or DRC) methods is communicated through the ADRC structure ^[36]. This is mostly because initially the DOBC structure was initially developed to deal with external disturbances of linear time-invariant systems, although the same structure equally applies to model uncertainties (internal disturbances) ^[32]. Still, the main difference between DOBC and ADRC lies on the different observers designed for the same lumped disturbance. By using the frame of the ADRC, the DOB in the DOBC can be generalised for estimating the total disturbance for nonlinear systems under certain conditions ^[82]. In that sense, ADRC is a more general category of DUEA methods, which can stabilise a more general class of non-linear converter systems, without setting strict mathematical constraints on the disturbances that can be estimated and rejected.

4. Comparison of Approaches for Enhancement of Robustness of Modern Feedback Control

The three approaches for the robustness enhancement present differences in their concept. Instead of adjusting the baseline feedback controller to the disturbances, similar to the adaptive control, in the DUEA approach the real converter plant is adjusted to what the feedback controller is designed for, i.e., the “disturbance-free” enforced plant (nominal converter model). Instead of trying to incorporate robustness objectives and disturbance models into the design of the baseline feedback controller, like in the robust control, the DUEA approach, tries to disregard the disturbances of the real converter plant for the feedback control design ^[32]^[36]. Therefore, in the DUEA approach, the focus of the control design for the converter is not on the accurate system identification or the high-fidelity converter and disturbance modelling, like in the model-based approaches of adaptive and robust control, but on the disturbance rejector design. There is a paradigm difference between the model-centric adaptive or robust control and the disturbance-centric DUEA approach ^[32].

With regard to the achieved disturbance rejection and the recovery of the nominal performance, the DUEA approach lies between the robust and the adaptive control. the DUEA approach might not manage to handle all disturbances appearing in the real converter plant. Therefore, it does not present so high robustness compared to the robust control, which is designed to achieve the highest possible robust stability and performance, by rejecting all defined disturbances. However, the DUEA approach presents promising inherent robustness too, thanks to the total disturbance estimation and rejection, and thus it can be regarded as a ``refined'' robust control approach \cite{Jun}. On the other hand, it can maintain the nominal performance, as the feedback controller is designed for the disturbance-free nominal converter model. This is opposite to the robust control, where the nominal performance is sacrificed for the robustness, due to the fixed conservative design according to the worst disturbance. In practice, this means that the converters, which operate, in most of their operational time, close to their nominal operating point, will usually present degraded performance when robust control is applied. In the case of the DUEA application, this issue does not occur. In addition, the robustness of the existing adaptive control techniques against model uncertainties is low, since the adaptive mechanism is highly converter-model-dependent. On the contrary, the DUEA techniques manage to reject such disturbances as part of the total disturbance. As a result, adaptive controllers might become unstable due to unmodelled high-order dynamics of the converter, whereas the DUEA techniques maintain good performance.

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