Abstract—This short paper illustrates the use of empirical gramians for controllability/observability analysis of nonlinear systems and compares the extracted information to results obtained from linear gramians and nonlinear observability matrices. It is shown that empirical gramians can more accurately represent controllability/observability of a nonlinear system over an operating region than if linear gramians are used. At the same time the information contained in empirical gramians is easier to extract and interpret than if Lie algebra-based controllability or observability matrices are used. I. INTRODUCTION mpirical controllability and observability gramians have been used for several years for model reduction of nonlinear systems [1]-[3]. However, no results of using empirical gramians for controllability and observability analysis have been reported in the literature. This work addresses this point by comparing observability information from linear gramians [4], lie-algebra-based observability matrices [5], and empirical gramians [1]. The presentation is limited to observability due to the allocated space; however, the results for controllability are similar in nature. II. PRELIMINARIES A. Linear Gramians For a linear system of the form Du Cx y Bu Ax x (1) the linear observability (W O,Linear ) and controllability (W C,Linear ) gramians [4] dt Ce C e W t A T t A Linear O T 0 , dt e BB e W t A T At Linear C T 0 , (2) Manuscript received September 10, 2004. A.K. Singh is a graduate student in the Department of Chemical Engineering, Texas A&M University, College Station, TX 77843 USA (email: abhay.singh@chemail.tamu.edu). J. Hahn is an assistant professor in the Department of Chemical Engineering, Texas A&M University, College Station, TX 77843 USA (phone: 979-8453568; fax: 979-8456446; e-mail: hahn@tamu.edu). can be used to compute observability and controllability of states of the system. B. Nonlinear Controllability and Observability Matrices For nonlinear systems ) ( ) ( ) ( x h y u x g x f x (3) it is possible to determine if a system is locally observable or controllable using concepts from differential geometry [5]. C. Empirical Gramians Empirical gramians [1], [2] p i r l s m ilm m C dt t rsc W 1 1 1 0 2 ) ( 1 r l s m T l lm l m O dt T t T rsc W 1 1 0 2 ) ( 1 (4) have been recently introduced for nonlinear systems of the form of equation (3). Refer to [2] for the exact definition of the variables from equation (4). III. EMPIRICAL GRAMIANS FOR CONTROLLABILITY/OBSERVABILITY ANALYSIS Empirical gramians have been used for model reduction of nonlinear systems and the benefit of using empirical gramians over conventional linear gramians for model reduction has been illustrated [2]. However, no current results exist in the literature that directly compare the information contained in empirical gramians to linear gramians or nonlinear geometric methods for controllability and observability analysis. An illustrative example is presented in this section and a comparison of the conclusions that can be drawn about observability from the available information is made. A. Illustrative Example Consider the following nonlinear system 2 2 2 1 2 2 1 1 1 1 1 1 x y x x x x x x (5) which has an equilibrium point at (x 1 ,x 2 ) = (1,1). An example without inputs is used due to the space constraint and, therefore, only observability of the system will be analyzed. The linearization of the system (5) at the equilibrium point is given (in deviation variables) by: On the Use of Empirical Gramians for Controllability and Observability Analysis Abhay K. Singh and Juergen Hahn E 2005 American Control Conference June 8-10, 2005. Portland, OR, USA 0-7803-9098-9/05/$25.00 ©2005 AACC WeA05.1 140