IEEE CONTROL SYSTEMS LETTERS, VOL. 3, NO. 4, OCTOBER 2019 823 Output-Constrained Robust Adaptive Control for Uncertain Nonlinear MIMO Systems With Unknown Control Directions Kapil Sachan and Radhakant Padhi Abstract —This letter proposes a new output-constrained robust adaptive controller for a class of uncertain multi- input multi-output nonlinear systems. In the adaptive con- trol synthesis, Nussbaum gain is introduced, which not only ensures the closed-loop stability of the system in presence of unknown control directions but also enforces the asymptotic tracking of the reference signal under non- parametric plant uncertainties. The output constraints are enforced by carrying out a novel transformation, which transforms the constrained system into an equivalent unconstrained system. It is proven that the closed-loop system is asymptotic stable in the sense of Lyapunov and the output of the system will remain bounded by the imposed output constraints. The effectiveness of the proposed control design is demonstrated through exten- sive simulations. Index Terms—Robust adaptive control, unknown control directions, Nussbaum function, constrained control. I. I NTRODUCTION W ITH the increase workplace complexities, where machines and human work together to attain optimal performance and productivity, the safety of both human, as well as machines, become a major concern. One of the ways to ensure safety is to impose hard constraints on the operational domain of the machines. This necessity opens up a whole new dimension in the theory of nonlinear control, where the con- troller is designed in such a way that the significant states of an uncertain plant remain bounded in a predefined domain. In the literature, barrier Lyapunov function (BLF) based adaptive backstepping control designs have been widely used to deal with system constraints [1]. These control schemes have been successfully utilized to control various classes of constrained nonlinear systems, such as strict-feedback single- input-single-output (SISO) nonlinear systems [1]–[3], multi- input-multi-output (MIMO) Euler-Lagrange systems [4]–[6] Manuscript received March 1, 2019; revised May 5, 2019; accepted May 24, 2019. Date of publication May 29, 2019; date of current version June 6, 2019. Recommended by Senior Editor J.-F. Zhang. (Corresponding author: Kapil Sachan.) The authors are with the Department of Aerospace Engineering, Indian Institute of Science, Bengaluru 560012, India (e-mail: kapil.121290@gmail.com; padhi@iisc.ac.in). Digital Object Identifier 10.1109/LCSYS.2019.2919814 and block triangular systems [7], [8]. However, due to the augmentation of backstepping technique, these control designs have several limitations, notably, the requirement of complex intermediate pseudo control variables or stabilizing functions and their derivatives in the control design, and the necessity of feasibility analysis before implementation. Consequently, many times these limitations result in conservative control design and satisfying these requirements can be a tedious task for a high dimensional system. Therefore, one of the focus of this letter is to develop a controller that preserve the benefits of BLF but overcome the drawbacks of backstepping control design. Handling non-parametric plant uncertainties along with unknown control directions or unknown control effectiveness matrix (CEM) is considered as another major challenge in adaptive control theory. To handle unknown control directions, output feedback sliding mode control (SMC) designs have been proposed using periodic or monitoring switching func- tions [9]–[12]. These asymptotically stable control designs are shown to be robust to model uncertainties and lead to fewer transients. However, it can be noticed that these SMC designs cannot handle constraints on the system, which is a primary objective of the current letter due to safety requirements. To cope up with both system constraints and unknown control directions, lately, state-feedback adaptive backstep- ping control designs, augmented with Nussbaum function and BLFs, are gaining a lot of attention ([2], [4], [5] and refer- ences therein). In these control formulations, BLFs are used to enforce the imposed system constraints. The Nussbaum function is used to learn unknown CEM, and universal approx- imators, such as neural networks (NNs) and fuzzy logic, are used to approximate the plant uncertainties [13]–[15]. However, along with the limitations of backstepping control philosophy, in the above control formulations, the CEM is either assumed as a unknown but positive definite matrix (for MIMO systems) or unknown scaler variable (for SISO systems). Thus, these controllers may not work for general nonlinear systems, where the CEM has unknown directions. Furthermore, for MIMO systems, the constraints are achieved only in norm sense and the domain of initial conditions are also conservative [4]. Thus a new control scheme is required for MIMO systems, which includes all the advantages of 2475-1456 c  2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.