Systems 2014, 2, 606-660; doi:10.3390/systems2040606 OPEN ACCESS systems ISSN 2079-8954 www.mdpi.com/journal/systems Article Adaptive Systems: History, Techniques, Problems, and Perspectives William S. Black 1;?, Poorya Haghi 2 and Kartik B. Ariyur 1 1 School of Mechanical Engineering, Purdue University, 585 Purdue Mall, West Lafayette, IN 47907, USA; E-Mail: [email protected] 2 Cymer LLC, 17075 Thornmint Court, San Diego, CA 92127, USA; E-Mail: [email protected] ? Author to whom correspondence should be addressed; E-Mail: [email protected]. External Editors: Gianfranco Minati and Eliano Pessa Received: 6 August 2014; in revised form: 15 September 2014 / Accepted: 17 October 2014 / Published: 11 November 2014 Abstract: We survey some of the rich history of control over the past century with a focus on the major milestones in adaptive systems. We review classic methods and examples in adaptive linear systems for both control and observation/identification. The focus is on linear plants to facilitate understanding, but we also provide the tools necessary for many classes of nonlinear systems. We discuss practical issues encountered in making these systems stable and robust with respect to additive and multiplicative uncertainties. We discuss various perspectives on adaptive systems and their role in various fields. Finally, we present some of the ongoing research and expose problems in the field of adaptive control. Keywords: systems; adaptation; control; identification; nonlinear 1. Introduction Any system—engineering, natural, biological, or social—is considered adaptive if it can maintain its performance, or survive in spite of large changes in its environment or in its own components. In contrast, small changes or small ranges of change in system structure or parameters can be treated as system uncertainty, which can be remedied in dynamic operation either by the static process of design or the design of feedback and feed-forward control systems. By systems, we mean those in the sense of classical mechanics. The knowledge of initial conditions and governing equations determines, Systems 2014, 2 607 in principle, the evolution of the system state or degrees of freedom (a rigid body for example has twelve states–three components each of position, velocity, orientation and angular velocity). All system performance, including survival or stability, is in principle expressible as functions or functionals of system state. The maintenance of such performance functions in the presence of large changes to either the system or its environment is termed adaptation in the control systems literature. Adaptation of a system, as in biological evolution, can be of two kinds–adapting the environment to maintain performance, and adapting itself to environmental changes. In all cases, adaptive systems are inherently nonlinear, as they possess parameters that are functions of their states. Thus, adaptive systems are simply a special class of nonlinear systems that measure their own performance, operating environment, and operating condition of components, and adapt their dynamics, or those of their operating environments to ensure that measured performance is close to targeted performance or specifications. The organization of the paper is as follows: Section2 surveys some of the rich history of adaptive systems over the last century, followed by Section3 with provides a tutorial on some of the more popular and common methods used in the field: Model Reference Adaptive Control, Adaptive Pole Placement, Adaptive Sliding Mode Control, and Extremum Seeking. Section4 provides a tutorial for the early adaptive identification methods of Kudva, Luders, and Narendra. A brief introductory discussion is provided for the non-minimal realizations used by Luders, Narendra, Kreisselmeier, Marino, and Tomei. Section5 discusses some of the weak points of control and identification methods such as nonlinear behavior, observability and controllability for nonlinear systems, stability, and robustness. This section also includes some of the solutions for handling these problems. Section6 discusses some of the interesting perspectives related to control, observation, and adaptation. Section7 presents some of the open problems and future work related to control and adaptation such as nonlinear regression, partial stability, non-autonomous systems, and averaging. 2. History of Adaptive Control and Identification The first notable and widespread use of ‘adaptive control’ was in the aerospace industry during the 1950s in an attempt to further the design of autopilots [1]. After the successful implementation of jet engines into aircraft, flight envelopes increased by large amounts and resulted in a wide range of operating conditions for a single aircraft. Flight envelopes grew even more with developing interest in hypersonic vehicles from the community. The existing autopilots at the time left much to be desired in the performance across the flight envelope, and engineers began experimenting with methods that would eventually lead to Model Reference Adaptive Control (MRAC). One of the earliest MRAC designs, developed by Whitaker [2,3], was used for flight control. During this time however, the notion of stability in the feedback loop and in adaptation was not well understood or as mature as today. Parks was one of the first to implement Lyapunov based adaptation into MRAC [4]. An immature theory coupled with bad and/or incomplete hardware configurations led to significant doubts and concerns in the adaptive control community, especially after the crash of the X-15. This caused a major, albeit necessary, detour from the problem of adaptation to focus on stability. The late 1950s and early 1960s saw the formulation of the state-space system representation as well as the use of Lyapunov stability for general control systems, by both Kalman and Bertram [5,6]. Aleksandr Systems 2014, 2 608 Lyapunov first published his book on stability in 1892, but the work went relatively unnoticed (at least outside of Russia) until this time. It has since been the main tool used for general system stability and adaptation law design. The first MRAC adaptation law based on Lyapunov design was published by Parks in 1966 [1]. During this time Filippov, Dubrovskii and Emelyanov were working on the adaptation of variable structure systems, more commonly known as sliding mode control [7]. Similar to Lyapunov’s method, sliding mode control had received little attention outside of Russia until researchers such as Utkin published translations as well as novel work on the subject [8]. Adaptive Pole Placement, often referred to as Self-Tuning Regulators, were also developed in the 1970s by Astrom and Egardt with many successful applications [9,10], with the added benefit of application to non-minimum phase systems. Adaptive identifiers/observers for LTI systems were another main focal point during this decade with numerous publications relating to model reference designs as well as additional stabilization problems associated with not having full state measurement [11–16]. However, Egardt [17] showed instability in adaptive control laws due to small disturbances which, along with other concerns such as instabilities due to: high gains, high frequencies, fast adaptation, and time-varying parameters, led to a focus on making adaptive control (and observation) robust in the 1980s. This led to the creation of Robust Adaptive Control law modifications such as: σ-modification [18], -modification [19], Parameter Projection [20], and Deadzone [21]. As an alternative for making systems more robust with relatively fast transients a resurgence in Sliding Mode Control and its adaptive counterpart was seen, particularly in the field of robotics [22–24]. The ideas of persistent excitation and sufficient richness were also formulated in response to the stability movement by Boyd, Sastry, Bai, and Shimkin [25–28]. These three decades were also a fertile time for nonlinear systems theory. Kalman published his work on controllability and observability for linear systems in the early 1960s, and it took about 10 years to extend these ideas to nonlinear systems through the use of Lie theory [29–44]. Feedback Linearization was formulated in the early to mid-1980s as a natural extension from applying Lie theory to control problems [45–52]. Significant improvements on our understanding of nonlinear systems and adaptation in the early 1990s was facilitated by the work on Backstepping and its adaptive counterpart by Kokotovic, Tsinias, Krstic, and Kanellakopoulos [53]. While Backstepping was being developed for matched and mismatched uncertainties, Yao and Tomizuka created a novel control method Adaptive Robust Control [54]. Rather than design an adaptive controller and include robustness later, Yao and Tomizuka proposed designing a robust controller first to guarantee transient performance to some error bound, and include parameter adaptation later using some of the methods developed in the 1980s. The previous work on nonlinear controller design also led to the first adaptive nonlinear observers during this time [55]. Another side of the story is related to Extremum Seeking control and Neural Networks, whose inception came earlier but development and widespread use as non-model and non-Lyapunov based adaptation methods took much longer. The first known appearance of Extremum Seeking (ES) in the literature was published by LeBlanc in 1922 [56]; well before the controls community was focused on adaptation. However, after the first few publications,