Automatica 42 (2006) 1611 – 1616 www.elsevier.com/locate/automatica Technical communique Mitigation of symmetry condition in positive realness for adaptive control  Itzhak Barkana a , ∗ , Marcelo C.M. Teixeira b , Liu Hsu c a Kulicke and Soffa Industries, Inc., 1005 Virginia Drive, Fort Washington, PA 19034, USA b Department of Electrical Engineering, FEIS/UNESP, Av. Brasil 56, 15385-000-Ilha Solteira-SP, Brazil c Department of Electrical Engineering, COPPE/UFRJ, P.O. Box 68504, 21945-970 Rio de Janeiro, RJ, Brazil Received 23 June 2005; received in revised form 16 May 2006; accepted 16 May 2006 Abstract Feasibility of nonlinear and adaptive control methodologies in multivariable linear time-invariant systems with state-space realization {A,B,C} is apparently limited by the standard strictly positive realness conditions that imply that the product CB must be positive definite symmetric. This paper expands the applicability of the strictly positive realness conditions used for the proofs of stability of adaptive control or control with uncertainty by showing that the not necessarily symmetric CB is only required to have a diagonal Jordan form and positive eigenvalues. The paper also shows that under the new condition any minimum-phase systems can be made strictly positive real via constant output feedback. The paper illustrates the usefulness of these extended properties with an adaptive control example.  2006 Elsevier Ltd. All rights reserved. Keywords: Control systems; Stability; Passivity; Uncertain systems; Adaptive control; Positive real systems; Almost strictly positive real (ASPR) 1. Introduction Consider the square system ˙ x(t) = Ax(t) + Bu(t), (1) y(t) = Cx(t). (2) Here, x is the n-dimensional state vector, u is the m-dimensional input vector and y is the m-dimensional output vector, and A, B , and C are matrices of corresponding dimensions. In var- ious methodologies of nonstationary control that use dynami- cal gains (Barkana & Kaufman, 1985; Fradkov, 1976; Sobel, Kaufman, & Mabus, 1982; Steinberg & Corless, 1985), the sta- bility analysis concerns both the state and the gains. Stability of the control system has been treated with positive definite (PD)  This paper has not been presented at any IFAC conference. This paper was recommended for publication in revised form by Associate Editor Gang Tao under the direction of Editor Miroslav Krstic. ∗ Corresponding author. Tel.: +1 215 784 6279. E-mail addresses: ibarkana@kns.com (I. Barkana), marcelo@dee.feis.unesp.br (M.C.M. Teixeira), liu@coep.ufrj.br (L. Hsu). 0005-1098/$ - see front matter  2006 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2006.05.013 quadratic Lyapunov functions of the form V(t) = x T (t)Px(t) + tr[(K(t) -  K) -1 (K(t) -  K) T ]. (3) Here, K(t) is the adaptive gain used with the controller u(t) = K(t)y(t) and  K represents an ideal output feedback gain. De- fine A K = A - B  KC. (4) Although the proofs of stability do not require the original sys- tem to be strictly positive real (SPR), they manage to prove stability of the controlled system only if there exists a constant output feedback gain  K (unknown and not needed for imple- mentation), such that the fictitious closed-loop system is SPR. The common state-space definition of the strictly positive real- ness property in linear time invariant systems is: Definition 1. A linear time-invariant system with a state-space realization {A K ,B,C}, where A K ∈ R n,n , B ∈ R n,m , C ∈ R m,n , with the m ∗ m transfer function T (s) = C(sI - A K ) -1 B , is called “strictly passive (SP)” and its transfer function (SPR) if there exist two positive definite symmetric (PDS) matrices, P and Q, such that the following two relations are