7. Interlude: A Look Around

Part of looking around is looking back. By reexamining the first six chapters of the book, one could notice that no literature references were provided alongside the presented information on ADRC. This was intentional on our end because we wanted to provide an introduction to ADRC without recurrent interruptions caused by referring to external materials. Now, as we are about to finish the first part of the book dedicated to the theoretical foundations of ADRC, it feels like a good time to review the so far presented material and give credit where credit is due. We named this entire chapter “Interlude” for a reason as we want to give the reader a bit of a break from the technical regime of the past few chapters and time to reflect on the things presented thus far. Here we want to give a literature overview and a historical perspective on ADRC for things we have shown in the first part. Usually placed at the beginning of a book, here such an overview is deliberately delayed as only now we have introduced enough information on ADRC for such a rundown to be meaningful to the readers. Therefore, let us now go through the chapters one by one. Before proceeding, we stress once again that even though we will be giving references to external materials, this book is written to be a self-contained introduction to ADRC, and you do not have to read anything else to get the gist of ADRC.

The phrase “look back” is understood here twofold. It is not only looking back at what was covered so far in the book by putting things in perspective and providing relevant bibliographical support but also looking back at the history of ADRC. To this end, we have been mostly focused on a specific variant of ADRC, one that uses a linear extended state observer and a linear state-space controller, in other words, a variant that we have been referring to as “LADRC.” However, it should be noted that the area of ADRC did not start with a linear version.

Tracking back the roots of ADRC, one can find that, as a concept, it was initially introduced by Prof. Jinqing Han as a nonlinear scheme [14], similar to the one we discussed in Sect. 6.3. ¹ It was only later streamlined through the linear variant [1]. To truly grasp the entirety of the long way ADRC has come, we recommend overview papers [23, 24], where the making of ADRC is described in detail. They offer a historical (and philosophical at times) perspective on the origins of ADRC, revisit how this paradigm came into being, and finally reexamine what helped ADRC to make the transition from an idea to an industrial technology. We intentionally leave to those articles the comprehensive telling of the story of ADRC until the point we pick it up. So when does our book enter this story which spans decades? And the answer is clear: 2003. This is the year when LADRC, with its convenient bandwidth parametrization-based tuning, was introduced in [1]. This is the point in time when we arrive and pick things up in the book. And finally, this is the variant of ADRC that we have been almost exclusively focused on so far in the book. But why this one? Are not there more powerful ADRC forms out there already?

One of the reasons behind such a choice is that the linear version allows a relatively simple introduction to the entire topic of ADRC and helps to form a solid base on which one can then build more advanced and more customized solutions. Our aim in writing this book was to put forward a text that develops along a single line of argument, hence the idea of using LADRC as a gateway to the world of ADRC seemed natural. This focused take helped us to tell a coherent story across Chaps. 2–6 and not worrying about distracting the reader with different variants or going off on a tangent with what else one can do with ADRC. It also allowed the reader to stay focused and understand the core concepts of the introduced methodology. Our objective is not to explore the full depth of mathematical completeness or cover all the nuances of ADRC but instead to give enough detail so that a reader can begin applying this approach as soon as possible. Focusing on a simple, fundamental form of ADRC thus helps to provide a sufficiently strong foundation, so that the reader can comfortably turn to the study of appropriate complementary literature on ADRC, and the LADRC variant facilitates that. The other reason behind focusing mostly on linear state-space ADRC in Part I, besides being straightforward, was shown in many documented instances to provide satisfactory results even for some complex control problems. Engineering practice is dominated by the linear variant. It is also from our own experience as we have been putting things to work with ADRC for years now. The undemanding structure and the limited number of tuning gains in LADRC paved the way for its widespread adoption. Therefore this variant fits well with the introductory style of this book.

It should be clear by now that, for reasons explained above, we focus in the book on a particular form of ADRC, which results from certain design choices we have made at the beginning, for example, in terms of choosing the plant model, disturbance model, observer structure, type of controller, and tuning approach. Now, we would like to take the opportunity to share some personal recommendations for further reading for those who wish to study ADRC in different aspects that go beyond this book. With this, we aim to broaden the readers’ horizons by showing what else the world of ADRC has to offer. Since the area of ADRC has been exponentially growing over the years, its current body of knowledge is massive. We had to thus apply certain criteria for the selection of materials we would like to recommend. In general, we focused on what else could you do in ADRC to arrive at roughly the same point as in this book. As a result, the below list provides literature support for things not covered in the book but potentially interesting after its completion. Furthermore, their selection is bound to involve compromise as we have not attempted to review all the results on ADRC that could be construed as being relevant.

Different Observer Types: Until this point in the book, we have mostly focused on the simple, linear, Luenberger-like ESO (except for the nonlinear variant mentioned in Sect. 6.3). But other observer structures may be used as well, each having its own set of advantages and disadvantages, as discussed in overview papers [25‐27]. For example, if one changes the total disturbance model in ADRC, then the observer design would naturally have to be aligned. Therefore, one can find in the literature extended linear forms such as generalized proportional integral observers, utilizing polynomial models for the total disturbance [28] or resonant ESOs, utilizing harmonic models [29‐31]. If one is interested in addressing specific aspects of the observer operation, there are specific structures that can accommodate, for example, those modifying the convergence rate by utilizing sliding modes [32], which in some cases can also offer finite-time convergence [33], or variants that can minimize the observer peaking phenomenon [34]. Furthermore, if certain signals related to the plant are available and can be utilized in the process of observer synthesis, then one could go for a reduced-order ESO to lower the overall complexity of the control system [35, 36]. There is also the idea of combining several observers within one ADRC design but is reserved for specific systems that can handle such increased computational burden. The multiobserver designs are commonly realized by connecting the observers in a cascade [37‐41] for extra denoising.

Different Controller Types: In the book, we have limited ourselves to the use of simple, classical controllers, like the ones seen in Chap. 3. Similar to the observer case, the reason behind such a choice was to keep the design and implementation simple and to stick with P and PD controllers, which can be found in any textbook on control. But in general, choosing the controller type in ADRC is up to the user. It is, however, a good engineering practice to introduce more advanced (and potentially more complicated) tools only if there is a clear practical need and justification for that. In [28], for example, the authors exploited the plant’s flatness property and designed high-order controllers called generalized proportional integral controllers. Another example can be found in [42], where the controller was designed using the sliding mode theory. There are many options to choose from related to how the controller is placed in the control system (example being a cascade control in [43]), what is the algorithm behind the calculation of the control signal (example being MPC in [41]), and to what specific system the ADRC is applied to (example being a controller tailored to high-order integral systems in [44]).

Different Tuning Methods: Without a doubt, the bandwidth parameterization tuning methodology has streamlined the development of ADRC. Considering its ease of use and reported satisfactory results in many cases, it is not surprising that it is the most common choice. We, therefore, rely on it throughout this book. But we also want to mention some of the options for choosing observer gains. Even though relatively little literature is available on alternative tuning methods, one can find some customized solutions, like characteristic ratio assignment to directly control the transient response [45] or those dedicated to time-delayed systems [46, 47]. One can also find in the literature some tuning approaches dedicated to very specific control problems that call for, for the price of increased computational complexity, more advanced tools like neural networks [48] or genetic algorithms [49].

Theoretical Results: Since we decided in the book to focus on the simple linear variant of ADRC, we can utilize classical linear systems theory, including pole analysis, to determine the stability of the control system. Those simple tools from the linear framework, taught in every control systems introductory course, apply to everything we have shown so far regarding LADRC. However, complex analysis, involving strong mathematical tools, is needed to verify more complex ADRC variants. The advanced theoretical results in ADRC mostly consider the study of observer convergence, stability of the closed-loop system, and robustness analysis (against parametric uncertainties, external disturbances, and unmodeled dynamics of the system) with papers [16, 50, 51] giving an overview of the current theoretical scene. One of the still challenging topics in the area of ADRC is establishing the applicability conditions for various classes of plants, with some progress made utilizing the tools of differential geometry [52, 53]. One of the key aspects of any ADRC design is the formulation of the total disturbance term, and works like [54] provide valuable insight into it.

Similar Control Methodologies: The idea of online estimation and rejection of total disturbance as opposed to relying on the necessity of having a detailed mathematical model of the controlled plant is not unique to ADRC. One can find other methodologies that handle control problems in such a way. Overview papers, like [55, 56], present the landscape of available approaches and the history of their evolution. The specific techniques mostly differ in the way how the aggregated uncertainty is being reconstructed. While in ADRC we use state observer for total disturbance estimation, in approaches like model-free control,² it is obtained with the means of algebraic estimation [57], in disturbance observers-based control schemes using properly constructed low-pass filters [58], or in balance-based adaptive controller using recursive least-squares procedure [59].

Equivalence with Other Control Methods: After the ADRC started to gain traction among the control community, a natural discussion started on how to position ADRC in the control landscape. Such discussion led to the research on finding equivalences with already existing control schemes. To this end, theoretical connections were found between ADRC and standard industrial controllers, like PI [60] and PID [61, 62] and also with more advanced control structures [63, 64]. Multiple researchers have shown independently that, with some simplification and for low-order plants, ADRC is indeed backward compatible with PI/PID. There is also now an interesting body of work trying to derive ADRC tuning rules from the PID parameters [65] or attempting to interpret classic controllers as disturbance observer-based structures [66]. A connection between error-based ADRC and classic industrial controllers was studied in [67]. Finding backward compatibility of ADRC, especially with PI and PID, is an important problem to solve as the ignorance of such an impediment arguably led to the stagnation of many of the advanced controllers and to the questioning of their relevance to engineering practice [68].

Other Books on ADRC: There are several books on ADRC on the market at the moment. They differ from each other in terms of focus and target audience. To enable a comparison with our work, we therefore want to give a brief overview. Our book is a concise, from scratch, step-by-step introduction to the world of ADRC where things are comprehensively explained from basic theory all the way to practical implementation. It stands on its own as a complete work. Hence, we view the following books as beneficial for those wishing to go beyond our book and extend their horizons by learning more about specific aspects of ADRC. We discuss them in the order of their publication year. In 2008, the first book on ADRC appeared and was published by the already mentioned Prof. Han [69]. In the book, written in Chinese, Han meticulously discussed all aspects of the original, nonlinear ADRC and its components. The book embraces the power of nonlinear feedback and puts it to full use through its incorporation in observer, controller, and tracking differentiator. The nonlinear ADRC is proposed there as a combination of both worlds, classical feedback control and modern control theory, which formulates a direct response to the limitations of PID. In 2014, a book was published, which offered a wider look at the topic of disturbance estimation and rejection [70]. The authors combined their extensive theoretical and practical knowledge and gave an overview of various techniques (not only ADRC) that can be used when the standard disturbance compensation via feedforward is not an option. The book not only contains the main concepts and design philosophies of various disturbance observer-based control approaches but also shows multiple simulation and experimental results. In 2016, a book was published that offered strong mathematical support for ADRC, including high-level convergence and stability proofs [71]. The focus of the book was on nonlinear systems and because of that covered a variety of advanced topics. The included theoretical derivations help to highlight the advantages of ADRC, including small overshooting, fast convergence, and energy savings. In 2017, two important books on ADRC were introduced to the market. The main focus of [72] was on exploring the relation between the development of information obtaining and processing technologies and the ability to merge various types of disturbances into one “equivalent” disturbance. The book also offers a summary of various developments in anti-disturbance control and focuses on descriptions of various control and filtering strategies. The second book [28] discusses ADRC as a methodology that differs from current robust feedback controllers, characterized by complex matrix manipulations, complex parameter adaptation schemes, and in other cases, induced high-frequency noises through the classical chattering phenomenon. The book does the above dissection using the notion of differential flatness. In 2021, a book on ADRC was published which had a channeled focus on a specific type of ADRC [73]. It investigated the theoretical efficiency and performance of so-called event-triggered ADRC, i.e., for nonuniform sampled-data control.

Software Support for ADRC Implementation: Since almost every commercial control equipment provides nowadays hardware or software PID function block supporting its fast and reliable implementation, some research was inevitably dedicated to facilitating the deployment of ADRC in industrial settings as well. In [74], the PLC-based implementation of ADRC was presented in the form of a general-purpose function block, allowing to choose the order of ADRC, switch between ESO and generalized proportional integral observer (with some user-defined order), and also switch between linear and nonlinear observer type (by properly setting the f_al(⋅) function parameters). In [75], a similar idea of a flexible function block for PLC was presented and offered some more practically oriented add-ons like derivative backoff and a built-in tuning approach based on the process step response.

Discrete-Time ADRC: A practical implementation of ADRC will almost always be in the form of software. Therefore, given the inherent discrete-time nature of the underlying target processor systems, it is necessary to provide discrete-time variants of ADRC for actual deployment in hardware. At the moment, readers interested in the use of discrete-time ADRC usually come across paper [76], in which the authors investigated and compared various digital implementations of the ESO. Over the years, there were also some customized ADRC solutions in discrete domain. For example, in the already mentioned book [73], the application of discrete-time ADRC was considered for the special case of event-triggered systems. In [77], the authors discretized the ESO using a discrete delta operator, which created a rapprochement between continuous and discrete dynamic system models and established a natural framework to investigate the behavior of discrete dynamic models in fast sampling limit. Surprisingly, the literature on discrete ADRC for uniform sampled-data control is quite scarce, given its crucial importance in practical implementation and deployment. This motivated us to work on this important subject and fill this gap. This is why we dedicate the entire upcoming second part of the book to this topic where we also present our results in that area.

Commercial Interest: The practically appealing high effectiveness-to-complexity ratio of ADRC has inevitably led to the first signs of increased interest in it from the industry. For example, Texas Instruments, a global semiconductor company, has incorporated the ADRC approach in some of its motion control products.³ Another example is the inclusion (since release R2022b) of an official ADRC library block in MATLAB. We will take a deeper look into this block in Appendix B.

Truth be told, all that we have shown so far in the book could be learned from scientific papers and books. We did, however, put our spin on things by offering our unique perspective (combining years of experience in academia and industry), structure, and way of presentation. But part of looking around is also looking forward. To this end, Part I, which dealt with fundamentals and continuous-time domain, is done, and now we are going practical with ADRC. And putting this into practice, as mentioned above, is really something different and can basically be done nowadays only in discrete time. Looking at the available literature, including the last couple of bullet points in the above further readings list, going into actual discrete implementation is not often done in terms of ADRC, and there are many open points when it comes to ironing out edges to facilitate practice-oriented implementation. Although there are many works documenting successful implementations of ADRC in practice, the literature investigating the foundations of ADRC in discrete domain seems underdeveloped. In our subjective estimation, more than 90% of ADRC-related papers either deal with its continuous-time and its original nonlinear variant or present only simulation results. There are a lot of materials on ADRC, some of which we collected here, but the air is very thin when it comes to discrete implementation and, as we saw earlier, the topic is undercovered in the ADRC landscape. This is where we see our main working field, both in our past works and in this book, as we wanted to contribute something in that area. This is also why the selling point of this book lies in its dual nature, in which necessary basics are first introduced in Part I, and only then practical implementation is covered in Part II, as it needs a separate, dedicated treatment. Concrete solutions for the application-oriented digital applications of ADRC in the discrete form are exactly what is yet to come in the book. We will need to, however, introduce a few more references in the second part of the book, but the style of reference will be different than in Part I as we will provide citations in the text whenever it is needed, not delaying it this time to the end of Part II. The second difference is that most of the cited references in Part II are our own works because we have been in this space and this has been our research interest for years.

Therefore, the second part of the book will be primarily based on our works [8, 78‐80]. Let us quickly break them down. In [8], to enable immediate comparability with existing classical control solutions and to support the adoption of ADRC in industrial practice, we have introduced a realizable transfer function implementation of continuous-time linear ADRC. In the same work, an exact implementation of discrete-time ADRC using transfer functions was introduced for the first time, with special emphasis on practical aspects such as computational efficiency, low parameter footprint, and windup protection. Then, in [78], we introduced an even more efficient implementation than the transfer function, which focused on minimizing the number of storage variables and algebraic operations. This “minimum-footprint implementation” remains the most efficient implementation to date. In [79], we have provided various options for discrete implementations of error-based ADRC, even with the same set of parameters as in the output-based variants. Finally, in [80], the proposed half-gain tuning results in similar closed-loop dynamics as the commonly employed bandwidth parameterization design, but with lower feedback gains, paving the way for using ADRC in more noise-affected applications. Although originally presented in continuous time, we will use it in Part II in its discrete form.

Before you transition to the second part of the book, be reminded once again that you do not need any of the above-cited works on discrete ADRC moving forward as we will provide all the necessary details. We want to keep the book self-contained also in the second part, so you can move through it without having to read any of the references mentioned in the upcoming chapters.