A Direct Model Reference Adaptive Control for SISO Linear Time Invariant Systems with Super-Twisting-like Terms Eder Guzm´ an and Jaime A. Moreno El´ ectrica y Computaci´ on, Instituto de Ingenier´ ıa, Universidad Nacional Aut´ onoma de M´ exico, 04510 M´ exico D.F., Mexico, {EGuzmanB, JMorenoP}@ii.unam.mx Abstract—Recently, the authors have proposed [2] a new Direct MRAC (Model Reference Adaptive Control) for SISO, relative degree 1, LTI systems, which has finite-time tracking and parameter convergence. These properties are obtained by adding some strong, i.e. non locally Lipschitz or discontinu- ous, nonlinearities to the controller and parameter estimation algorithm of the classical MRAC. The Reference Model (RM) in [2] has been restricted to a SPR transfer function with no zeroes. The objective of this paper is to show that the same algorithm used in [2] is valid for any RM, having an SPR transfer function with relative degree one. However, the zero dynamics of the RM precludes the finite time convergence, but the proposed algorithm still converges faster than the classical one. A Lyapunov-based approach is used to prove these properties. Some simulations illustrate how the proposed algorithm provides the MRAC with a much better convergence than the classical algorithm, even in the presence of noise, for RMs with zero dynamics. I. INTRODUCTION The direct Model Reference Adaptive Control (MRAC) is a well-known approach for adaptive control of linear and some nonlinear systems [14], [17], [7], [3]. The objective of this kind of control is that the output plant y p tracks a desired output model y m by means of a controller whose parameters are tuned by an adjustment mechanism (adaptive law). When the Linear Time-Invariant (LTI) plant has relative degree n ∗ = 1 and the reference model is Strictly Positive Real (SPR), the controller is particularly simple to implement, and to design. In [2] a new MRAC capable of having finite-time convergence of the tracking error and the parameters under appropriate Persistence of Excitation Conditions is presented. This algorithm also enhances the robust properties, but the analysis was restricted to a class of Reference Models (RM) with no zeros, i.e. the numerator of the RM’s transfer function is a polynomial of degree zero. This new MRAC is based on a new recursive, finite-time parameter estimation algorithm presented in [13], which resembles the classical parameter estimation algorithms, but with extra strong (not locally Lipschitz or discontinuous) nonlinear terms added, so that the convergence and robustness properties of the classical algorithm are enhanced. These nonlinear terms are borrowed from the Super-Twisting Algorithm (STA), a second-order sliding mode algorithm proposed for the first time by [4]. Due to its strong convergence and robustness properties, this algorithm has proved to be useful in sev- eral applications as, for example, exact differentiators [5], [6], output feedback controllers [6], and observers [1]. A Lyapunov function for this algorithm has been presented in [9] (see also [10]), and the algorithm has been generalized in [12], [11]. The parameter estimation algorithm proposed in [13] inherits some of the properties of the Generalized Super-twisting Algorithms. The objective of this paper is to prove that the RM’s transfer function can have in the numerator a polynomial of any degree, as long as the relative degree 1 is maintained. However, the zero dynamics of the RM precludes the finite- time convergence of the algorithm. In spite of this, the pro- posed algorithm shows excellent convergence and robustness properties. A Lyapunov-based approach is used to prove the former and to relax the assumption on the RM. II. MRAC WITH RELATIVE DEGREE n ∗ = 1 A. Classical MRAC The control in the direct MRAC has an structure that depends on unknown constant parameters which are updated by an adaptive law. This kind of design allows to manipulate the control so that the stability analysis of the error equation can be made through SPR-Lyapunov approach in order to generate the mentioned adaptive law. For the classical Direct MRAC [14], [7], [3] one considers the SISO, LTI plant ˙ x p = A p x p + B p u p y p = C T p x p (1) where x p ∈ ℝ n ; y p , u p ∈ ℝ 1 and A p , B p , C p have the appropri- ate dimensions, that can also be described in the input/output form y p = G p (s)u p = k p Z p (s) R p (s) u p (2) where G p (s) is the transfer function, Z p , R p are monic polynomials and k p is a constant referred to as the “high frequency gain”. It will be assumed that the plant has relative degree n ∗ = 1. A reference model, selected by the designer, is described by ˙ x m = A m x m + B m r , x m (0)= x m0 y m = C T m x m (3) where x m ∈ ℝ p m for some integer p m ; y m , r ∈ ℝ 1 and r is the reference input, which is assumed to be uniformly 12 th IEEE Workshop on Variable Structure Systems, VSS’12, January 12-14, Mumbai, 2012 978-1-4577-2067-3/12/$26.00 c 2011 IEEE 337