1180 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS—PART B: CYBERNETICS, VOL. 39, NO. 5, OCTOBER 2009 Approximate Adaptive Output Feedback Stabilization via Passivation of MIMO Uncertain Systems Using Neural Networks Artemis K. Kostarigka and George A. Rovithakis, Senior Member, IEEE Abstract—An adaptive output feedback neural network con- troller is designed, which is capable of rendering affine-in-the- control uncertain multi-input–multi-output nonlinear systems strictly passive with respect to an appropriately defined set. Con- sequently, a simple output feedback is employed to stabilize the system. The controlled system need not be in normal form or have a well-defined relative degree. Without requiring a zero-state detectability assumption, uniform ultimate boundedness, with re- spect to an arbitrarily small set, of both the system’s state and the output is guaranteed, along with boundedness of all other signals in the closed loop. To effectively avoid possible division by zero, the proposed adaptive controller is of switching type. However, its continuity is guaranteed, thus alleviating drawbacks connected to existence of solutions and chattering phenomena. Simulations illustrate the approach. Index Terms—Neurocontrol, output feedback, passivation. I. I NTRODUCTION T HE USE OF passivity in systems theory has a long his- tory. Since the establishment of the relationship between passivity and Lyapunov stability by Willems [1], [2], scientists have been extensively using this powerful tool in a variety of nonlinear control problems. Asymptotic stabilization, even by pure linear feedback, is a significant property of passive systems. The desire to exploit passive system’s inherent properties gave birth to the feedback passivation approach, according to which a system is rendered passive with the use of a feedback control law. First insight was given by Byrnes et al. in [3], where necessary and sufficient conditions for state feedback equivalence to a passive system were provided. These con- ditions required weakly minimum phase systems, possessing a relative degree that is equal to one. In addition, with the aid of a zero-state detectability assumption, global asymptotic stabilization could be guaranteed. Sufficient conditions for the output feedback passivation problem were introduced in [4], while necessary and suffi- cient conditions for the output feedback exponential passivation problem were reported in [5]. Output feedback equivalence to an incrementally passive system was discussed in [6]. Sub- sequent efforts [7], [8] were dedicated toward relaxing the Manuscript received May 27, 2008; revised October 22, 2008. First pub- lished March 24, 2009; current version published September 16, 2009. This paper was recommended by Associate Editor H. Gao. The authors are with the Department of Electrical and Computer Engineer- ing, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece. Digital Object Identifier 10.1109/TSMCB.2009.2013477 relative-degree-one restriction. Attempts to remove the zero- state detectability assumption were provided in [9], for the case of full-state feedback, through coordinated passivation designs. All aforementioned works require at least perfect knowledge of a nominal system. When the controlled system is considered unknown, existing control methodologies do not apply. Owing to the large system uncertainties, the problem of exact feedback passivation is intractable and is thus practically substituted by the concept of feedback passivation with respect to a set, meaning that a system can be rendered passive via a feed- back whenever its output belongs to the complementary of an arbitrarily small set. In this respect, zero-state detectability proves insufficient to establish system stabilization. In [10] and for the special case of nonnegative nonlinear systems, neural networks and the results of [5] were combined to guarantee output feedback stabilization through exponential passivation. In this paper, we follow a two-phase control design proce- dure. In the first phase, an adaptive output feedback scheme is proposed, forming an inner control loop, which is capable of rendering an affine-in-the-control multi-input–multi-output (MIMO) nonlinear system, possessing unknown nonlinearities, equivalent to a strictly passive one with respect to an appro- priately defined set. In the second phase, an outer loop is formulated via a simple output feedback to guarantee a uniform ultimate boundedness property, with respect to arbitrarily small sets, of both the system output and the state. Furthermore, all other signals in the closed loop are kept bounded. To overcome the high amount of system’s uncertainty, we ex- ploit the approximation capabilities of neural networks, which have been proven very efficient in nonlinear system identifica- tion and control [11]–[31]. To avoid the possibility of division by zero (a highly sig- nificant and frequently appearing problem, particularly within approximation-based adaptive control designs), the inner con- trol loop is of switching type. However, efforts have been devoted to guarantee its continuity, thus alleviating problems related to existence of solutions and chattering phenomena. Other than the aforementioned information, the results of this paper extend the currently available ones in the following ways: 1) A new concept (passivation with respect to a set) is introduced to help deal with the affine-in-the-control nonlinear systems, possessing unknown nonlinearities; 2) no relative- degree-one assumption is required; and 3) we do not dwell on a zero-state detectability condition to prove output feedback stabilization. 1083-4419/$25.00 © 2009 IEEE