Special issue on PID control in the information age: Theoretical advances and applications

Abstract

For well over a century, from the enabling of early governors to a standard tool in process control, PID (proportional–integral–derivative) control stood out as the most favored method for its unparalleled simplicity, ease of implementation, and cost-effectiveness, and indeed has been an enduring gift to feedback control. For a technique so mature and seemingly so ageless, what can be new and worthy of attention? Surprisingly, even in a post-modern and information-centric era of time, PID control continues to demonstrate its sustained power and inexplicable charm, with incredible vitality and widespread acceptance by industrial control and automation communities. In coping with the increased complexity and challenge of contemporary engineering systems, there has been renewed interest in PID control, on fundamental issues concerning robustness, performance, structure, and optimization, as well as on practical design, implementation, and tuning. Recent studies on PID control have led to new and improved design and tuning rules, and expansions into new, emerging problem areas and application domains. These call for a reexamination and further development of the fascinating subject of PID control theory and applications. This special issue responds to the recent surge of research activities and is aimed at bringing together the latest advances central to PID control. A total of 22 articles are featured in the issue, which attests to the richness and versatility of PID control, and its depth and breadth. We must marvel at the fact that even for such a century-old topic, there are still gems to be discovered. Exactly one hundred years ago, in the year of 1922, Nicolas Minorsky published his landmark paper—believed to be the first published theory—on PID control, “Directional Stability of Automatically Steered Bodies” (Journal of the American Society of Naval Engineers, vol. 34, no. 2, May 1922, pp. 280–309), thus heralding in the birth of the formal PID control theory, alongside the theories of Maxwell, Routh, and Hurwitz. This, we take as a pleasant coincidence, and it appears fitting for commemoration. Thus, at its centennial mark, we are pleased to present this special issue and hope that it will inspire new thoughts and ideas for the continuing development and prosperity of PID control in years to come. It should be said that although intended for the reader's convenience this classification is rather crude, and is by no means accurate nor necessary; in fact, each article may be intertwined with some others from a different group, and hence may well cross into several categories. We briefly introduce below the contributions. Traditionally, PID control presumes, except in rare instances, a linear plant model, while in physical implementation, the underlying system contains intrinsically nonlinear characteristics. As such, PID design and tuning techniques for nonlinear systems are of both theoretical interest and practical importance. Zhao and Guo1 considered a regulation problem for a class of non-affine nonlinear uncertain plants. They constructed an extended PD (EPD) controller and showed that by tuning the EPD parameters appropriately from a set defined by the nonlinear characteristics, it is possible to achieve global regulation. Lyu and Lin2 studied the PID control problem for planar uncertain nonlinear systems with a bounded time-varying input delay. The system uncertainty is characterized by bounds on the growth rates of the nonlinear function and by a bound on the range of time delay in the system model. The authors obtained linear matrix inequality (LMI) conditions for the uncertain nonlinear system to be stabilized globally uniformly asymptotically under PID feedback, and a bound on the delay range. Similarly, Xiong and Hou3 investigated uncertain nonlinear non-affine discrete-time systems. By employing a dynamic linearization scheme, they showed that a PID controller based on the linearized model, when appropriately constructed, can stabilize asymptotically the uncertain nonlinear system. PID control is historically a somewhat empirical method that relies heavily on trial-and-error tuning, requiring no explicit model of the plant. Following the trend of recent theoretical investigations of PID control, this set of articles seeks to develop a fundamental understanding of properties of PID controllers. Under circumstances where partial or full knowledge of the plant is known (e.g., the order, pole, and delay range), such understanding can aid the design and tuning for the betterment of system robustness and performance. Özbay and Gündes4 addressed the problems of pole placement and delay margin by the design of low-order controllers, including specifically PID controllers. Here the delay margin is referred to as the largest range of delay permissible so that the delay plant can be stabilized by a fixed controller over that range. For second-order unstable plants, their method results in a PID controller that guarantees a desired level of delay margin and meets pole placement conditions. The authors also showed that the method can be extended to plants containing a chain of integrators. Méndez-Barrios et al.5 addressed the ill-posedness issue that may exist with the implementation of PID controllers, which may arise when the derivative control is approximated by a delay-difference operator. The authors analyzed the asymptotic behavior of the system characteristic roots, with which they characterized cases when delay-difference approximation does lead to an improperly posed stability problem. Otherwise, when the approximation is properly posed, algorithms are provided to compute bounds on the allowable delay and a robustness measure of the controller. Gu et al.6 presented a geometrical description for the set of all stabilizing PID controllers. This set is determined by the stability crossing surface, where the system becomes marginally stable. The authors draw upon concepts from differential geometry and showed that the crossing surface is completely characterized by a curve known as the discriminant of the surface. With this characterization, a method is then developed to identify the regions of the PID parameters such that the system is stabilized. Liu et al.7 explored PD controller design for positive linear systems. The goal is to design a PD controller, whose proportional and derivative gains may undergo variations, so that the closed-loop system is stable and remains positive. Necessary and sufficient conditions are given in terms of a linear program or LMIs. Despite its many advantages, PID control is however limited and can be ill-equipped to cope with high-order dynamical systems, due to its limited degree of design freedom, which boils down to the tuning of only three control parameters. Such situations arise when, for example, the plant has complex dynamics or significant disturbances may take place. Where necessary or beneficial, it is desirable to extend and augment the PID structure. More pointed tuning rules tailored to specific applications can also be useful. Fliess and Join8 addressed the limitation and proposed an alternative model-free control architecture for reference tracking, which essentially consists of a PID controller and additionally an estimator of the plant, taking into consideration of the reference trajectory. Laboratory experiments were conducted to show the effectiveness of this augmented architecture. Nie et al.9 considered another similar, filter-augmented control architecture, known as the active disturbance rejection control (ADRC). They suggest that the well-known Ziegler–Nichols tuning rules can be incorporated into the ADRC scheme. More generally, the ADRC architecture is investigated by Zhong et al.10 for uncertain multi-input multi-output nonlinear non-affine systems, where the authors showed that when tuned using ADRC rules, PID control can achieve satisfactory transient and steady-state tracking performance, and that an ADRC-assisted PID control can tolerate nonlinear uncertainties, thus establishing to a certain degree the robustness of PID controllers. Shi et al.11 considered a so-called dynamic equational PID controller, which augments the derivative control in the PID controller to high-order differentials. The scheme is applied to the control of uncertain systems with possible parameter variations in bounded intervals, and the system robustness is evaluated in a probabilistic sense. Experiments on, for example, a water tank system are employed to validate the method. On the other hand, Bahavarnia and Tavazoei12 proposed a probabilistic framework for, for example, PI control of uncertain nonlinear systems, where the system parametric uncertainties are modeled based on probability distributions and the PI controller is tuned based on centroids defined by the probability distribution and a prescribed parameter set. Zhang and Fridman13 considered sampled-data implementation of an extended, “high-order” PID controller, where the derivative control action also utilizes high-order differentials of the system output response, which in turn are approximated by finite differences of the output response sampled at sampling instants. LMI conditions are provided for the sample-data extended PID controller to stabilize the continuous-time plant, and an event-triggered condition is derived for the purpose of reducing the number of samples required for meeting the stabilization condition. Yet drawing upon reinforcement learning theory, Yu et al.14 took one step further, to integrate PID control with soft actor-critic (SAC) algorithms, which are much scrutinized in the recent machine learning and control literature. A two-layer hierarchical control structure is then proposed, which includes an upper-level controller based on SAC, and a lower-level controller as a PID controller. It is argued that with the SAC layer in place, the control design can be model-free, and the SAC–PID combination has the merit of robustness with respect to different systems. Multi-agent consensus and cooperative control are a popular contemporary research front of systems and control in recent years. It is thus natural and plausible to explore how in the presence of added network constraints, PID consensus protocol as a simple means of feedback may help enable and facilitate consensus attainment. Zhang et al.15 addressed the performance and robustness of first-order multi-agent systems under PI consensus protocol. Network sensitivity and complementary sensitivity functions are introduced to characterize the system performance. Sensitivity and complementary sensitivity minimization problems, in weighted and unweighted forms, are solved by optimizing the PI parameters, which lead to conditions for consensus robustness and performance robustness. Similarly, Rong et al.16 also considered the robust consensus problem for integrator agents under distributed PI protocols over an uncertain network, where the network uncertainty is modeled by an unknown transfer function bounded in the H∞ norm. The authors also introduced a generalized network complementary sensitivity function and showed that a necessary and sufficient condition for robust consensus can be determined by computing the H∞ norm of the generalized network complementary sensitivity function. Ma et al.17 investigated consensus robustness with respect to variable network delays, where robustness is measured by the delay consensus margin (DCM), that is, the largest range of delay allowable to ensure consensus attainment in spite of variations in the delay. They considered second-order multi-agent systems under distributed PD feedback, alternatively known as position and velocity feedback protocols, and showed that the exact delay consensus margin can be obtained by solving a univariate quasi-concave optimization problem. The latter problem can generally be solved using convex optimization or gradient-based numerical methods. Furthermore, Nan et al.18 present a study of bipartite consensus problems over directed communication graphs, which concern cooperative-antagonistic networks. The authors proposed a distributed adaptive PI-gain protocol, which is shown to be capable of achieving bipartite consensus. When extended to observer-based protocols, the PI-gain protocol design is shown to be extendable to solve the bipartite consensus problem with nonlinear agents. The application-oriented studies of this special issue are focused on mechanical systems and mechatronic systems, which is yet another field where PID control has seen extraordinary successes, for reasons ranging from physical constraints to the necessity to provide cost-effective solutions (in the control of, e.g., robotic manipulators, motor drives, disk drives, and electronic chips). Kovacs and Insperger19 studied the PID control of an inverted pendulum, which is intended for modeling human balancing task such as stick balancing on the fingertip. They analyzed a particular variant of PID control, that is, the delayed proportional–derivative–acceleration (PDA) feedback, together with predictor feedback control. Parametric uncertainties are considered in the form of perturbed feedback gains, interpreted as sensor errors, and the system performance is measured by the critical length, that is, the length of the shortest pendulum that can be balanced. Robust stabilizability analysis using the real structured stability radius suggests that PD and PDA controllers are potentially limited in comparison to predictor feedback; the latter can balance pendulum of a shorter length. Cui et al.20 investigated the prescribed-time tracking control problem for Euler–Lagrangian systems in the presence of modeling uncertainties and time-varying state constraints. They showed that by using PI control schemes with time-varying adaptive tuning gains, among other benefits, prescribed tracking precision can be achieved in finite time, while ensuring that state constraints be satisfied. Qiao et al.21 developed adaptive PID controllers for trajectory tracking of robotic manipulators with uncertainties, which consist of a standard PID controller augmented by an adaptation law. It is shown that the adaptive PID controller is robust with respect to the uncertainties and adaptive with reference to the unknown manipulator and load parameters. It is also shown that the tracking errors for both the manipulator joint position and velocity converge asymptotically to zero. Finally, Gao et al.22 present a robust adaptive, fault-tolerant control scheme for six degrees of freedom (DOF) unmanned aerial vehicles (UAVs) in the presence of modeling uncertainties and actuation faults. The control scheme is comprised of a PD controller with adaptively tuned proportional and derivative gains, which also considers implementation errors. The authors showed that the controller is capable of tolerating actuation faults while maintaining stability and performance requirements. We would like to thank the authors for contributing their work to this special issue and the reviewers for their effort devoted to the review process, which together ensures the quality and timely reviews of the articles. We also want to thank the Editor-in-Chief, Professor Michael Grimble, and the Wiley editorial team under his leadership, whose diligence ensures the timely publication of this issue.