Mini Review Volume 3 Issue 5 - October 2018 DOI: 10.19080/RAEJ.2018.03.555623 Robot Autom Eng J Copyright © All rights are reserved by József K Tar On the Alternatives of Lyapunov’s Direct Method in Adaptive Control Design Hamza Khan, József K Tar* and Imre J Rudas Óbuda University, Hungary Submission: October 14, 2018; Published: October 26, 2018 *Corresponding author: József K Tar, Óbuda University, Hungary; Tel: ; Email: Robot Autom Eng J 3(5): RAEJ.MS.ID.5555623 (2018) 00136 Introduction There is a wide class of model-based control approaches in which the available approximate dynamic model of the system to be controlled is “directly used” without “being inserted” into the mathematical framework of “Optimal Control” (OC). A classical example is the “Computed Torque Control” (CTC) for robots [1]. However, in the practice we have to cope with the problem of the imprecision (very often incompleteness) of the available system models (in robotics e.g. [1,2], modeling friction phenomena e.g. [3- 7], in life sciences as modeling the glucose-insulin dynamics e.g. [8- 11] or in anesthesia control e.g. [12-14]). Modeling such engines as aircraft turbojet motors is a quite complicated task that may need multiple model approach [15-18]. Further practical problem is the existence and the consequences of unknown and unpredictable “external disturbances”. A possible way of coping with these practical difficulties is designing “Adaptive Controllers” (AC) that somehow are able to observe and correct at least the effects of the modeling imprecisions by “learning”. Depending on the above available information on the model various adaptive methods can be elaborated. If we have precise information on the kinematics of a robot and only approximate information is available on the mass distribution of a robot arm made of rigid links the exact model parameters can be learned as in the case of the “Adaptive Inverse Dynamics” (AID) and the “Slotine-Li Adaptive Controller” (SLAC) for robots that are the direct adaptive extensions of the CTC control. An alternative approach is the adaptive modification of the feedback gains or terms [19]. The “Model Reference Adaptive Control” (MRAC) has double “intent”: a) it has to provide precise trajectory tracking, and b) for an outer, kinematics-based control loop they have to provide an illusion that instead of the actually controlled system, a so called “reference system” is under control (e.g. [20-22]). The traditional approaches in controller design for strongly nonlinear systems are based on the PhD thesis by Lyapunov [23] that later was translated to Western languages (e.g. [24]). (In this context “strong nonlinearity” means that the use of a “linearized system model” in the vicinity of some “working point” Abstract The prevailing methodology in designing adaptive controllers for strongly nonlinear systems is based on Lyapunov’s PhD Thesis he defended in 1892 to study the stability of motion of systems for the solution of the equations of motion of which no closed form analytical solutions exist. The adaptive robot controllers developed in the nineties of the 20thcentury guarantee global (often asymptotic) stability of the controlled system by using his ingenious Direct Method that introduces a Lyapunov function for the behavior of which relatively simple numerical limitations have to be proved. Though for various problem classes typical Lyapunov function candidates are available, the application of this method requires far more knowledge than the implementation of some algorithm. Besides requiring creative designer’s abilities, it often requires too much because it works with satisfactory conditionsinstead of necessary and satisfactoryones. To evade these difficulties, based on the firm mathematical background of constructing convergent iterative sequences by contractive maps in Banach spaces, an alternative of Lyapunov’s technique was so introduced for digital controllers in 2008 that during one control cycle only one step of the required iteration was done. Besides its simplicity the main advantage of this approach was the possible evasion of complete state estimation that normally is required in the Lyapunov function- based design. Though the convergence of the control sequence can be guaranteed only within a bounded basin, this approach seems to have considerable advantages. In the paper the current state of the art of this approach is briefly summarized. Keywords: Adaptive control; Lyapunov function; Banach space; Fixed point lteration Abbreviations: AC: Adaptive Control; AFC: Acceleration Feedback Controller; AID: Adaptive Inverse Dynamics Controller; CTC: Computed Torque Control; FPI: Fixed Point Iteration; MRAC: Model Reference Adaptive Control; OC: Optimal Control; PID:Proportional, Integrated, Derivative; RARC: Resolved Acceleration Rate Control; RHC: Receding Horizon Controller;SLAC: Slotine-Li Adaptive Controller;