JOURNAL OF GUIDANCE,CONTROL, AND DYNAMICS Vol. 24, No. 3, May– June 2001 Control of Linear Second-Order Systems by Fuzzy Logic-Based Algorithm Kelly Cohen, ¤ Tanchum Weller, † and Joseph Z. Ben Asher ‡ Technion—Israel Institute of Technology, 32000 Haifa, Israel The control of exible structures employing the passivity approach has been extended to systems having non- collocated input/output pairs by introducing an observer that incorporates the nominal dynamical model of the plant. The passive observer-based control is applied to the American Control Conference benchmark problem, whereby, the control force emulates a dynamic vibration absorber attached to a virtual wall with passive control elements (spring, mass, and dashpot). The springs and mass elements of the controller are constant, whereas the damping coef cients are selected as time dependent in an attempt to choose continuously the most appropriate amount of damping in compliance with the design goals. A novel approach is introduced, whereby the passive observer-based control law is modi ed by varying the damping coef cient of the virtual dashpot by means of an adaptive fuzzy logic algorithm. This modi ed system exhibits quick settling times and desirable performance characteristics. Results from the statistical robustness analysis for the developed controller are compared to 10 other (linear) solutions of the benchmark problem. The comparison is based on robust stability, robust performance (settling time), and control effort. The results obtained by the adaptive fuzzy logic algorithm are superior to those obtained by all other methods, and, consequently, further application of the fuzzy algorithm is advocated. I. Introduction D URING the years 1990– 1992, certain benchmark problems forrobustcontroldesignwere presentedat the American Con- trol Conference (ACC). One of these problems, referred to by Wie and Bernstein 1 as ACC benchmark problem 1, was concerned with vibration control of a two-mass system with an uncertain spring constant in view of a transient disturbance (Fig. 1). The simplicity of this problem provided a transparency that enabled it to be an in- teresting tool for comparison of a variety of robust control design methodologies. Nevertheless, this problem is nontrivial because it couples both rigid- and exible-body modes with plant uncertainty and noncollocatedsensor and actuator. In addition, sensor readings are contaminated by a high-frequencysensor noise. The ACC benchmark problem may be classi ed as a exible structurerepresentedby a second-orderdynamicsystem.For such a class of dynamic systems, Juang and Phan 2 presented a robust con- trollerthatisa passivedesignbasedon virtualsecond-orderdynamic system comprising virtual mass, spring, and dashpot elements. The virtual mechanisms incorporated into the passive design serve only to transferand dissipatethe energy of the system, therebymaintain- ing the stability of the system. In addition, Juang and Phan 2 showed that overall closed-loop stability was guaranteed, independentlyof the system structural uncertainty and perturbationsin the temporal plant dynamics. These second-order controllers may also be termed collocated, and they consistof compatiblepairs of actuatorsand sensors,which maybedistributedthroughoutthestructure. 3 However, in many real- life situationsinvolvingthecontrolof exiblestructures,becauseof physical placement and hardware limitations, absolute collocation may sometimes be impossible. 4 Furthermore, for the case of non- collocated actuator and sensor pairs, such as the ACC benchmark problem,strictlypassivefeedbackno longerguaranteesstability.To circumvent this problem, Hughes and Wu 5 presented an observer- based extension of the passive controller design for the described noncollocated case. This approach, based on the nominal dynamic modelof the system,is a directgeneralizationof the dissipativecon- troller, whereby the passive output is synthesized using an observer Received 5 November 1999; revision received 1 August 2000; accepted for publication8 September 2000.Copyright c ° 2000 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. ¤ Graduate Student, Department of Aerospace Engineering. † Dean, Department of Aerospace Engineering. ‡ Associate Professor, Department of Aerospace Engineering. as opposed to the availabilityof physical measurement as required. This generalization,which includesapplicationto the noncollocated case is, however,at the expenseof two of the inherentcharacteristics of passive controllers, the sacri ce of model independenceand the taking of stability robustnessfor granted. In this paper, a second-order passive controller is applied to the ACC benchmark problem whereby the control force emulates a virtual dynamic vibration absorber attached to a virtual wall by means of a virtual spring (Fig. 2). The springs and mass elements of the controller are tuned to introduce two virtual low-frequency exible modes instead of the single rigid-bodymode. Furthermore, a control law, which is based on the principlesof fuzzy logic control is introduced, to tune the damping parameter continuously of the earlier-describedpassive controller. The main advantages of using a fuzzy approach are the relative ease and simplicity of implementation and the robustness charac- teristics. The parameters of the described absorber may be adapted to provide fairly fast control for large deviations, of the measured state of the plant from the desired state, and a minor amount of control for small deviations. The successful implementation of a fuzzy logic controller depends, among other design aspects, on the heuristicrule basefrom whichcontrolactionsare derived.To obtain the required heuristic physically based insight, a single-degree-of- freedom (DOF) system based on optimal control theory will be ex- aminedanalyticallytoobservethecharacteristicsofaminimumtime solution.Later,a fuzzylogicnonlinearmappingfunction,which has the potential of being a universal approximator, 6 is applied to emu- late the minimum-time solution. The resulting rule base is the core of the controllaw thatis appliedhereinto the ACC controlproblem. Finally, we examine whether the closed-loopsystem should pro- vide satisfactorystability and performance characteristicsnot only forthenominalplantbutalsoovertherangeofvaluesassociatedwith parameter uncertainties.As mentioned by Wie and Bernstein, 1 the feedbackcontrollershoulddisplayreasonableperformance/stability robustness.To this end, Stengel and Marrison 7 presentedsome eval- uation criteria that concern the selection of appropriatemeasures of robustness of the ACC benchmark problem and that demonstrate that the described evaluation criteria may be satis ed by the appli- cation of stochastic robustness analysis (SRA). SRA 7,8 involvesdeterminingtheprobabilityofunsatisfactorysta- bility or performanceresultingfrom expectedparameterunstability. Furthermore, SRA was shown to provide a useful, unifying ana- lytical framework that is intuitive and to have a direct, physical meaning. The de nitions and principles of the SRA adhered to in 494