Riju Samanta et al. Int. Journal of Engineering Research and Application www.ijera.com ISSN : 2248-9622, Vol. 3, Issue 5, Sep-Oct 2013, pp.741-744 www.ijera.com 741 | Page Design of Full Order Observer Using Generalized Matrix Inverse for Linear Time Invariant Systems Riju Samanta 1 , Avijit Banerjee 2 , Gourhari Das 3 1 (Department of Electrical Engineering (P.G., 2014 Control System), Jadavpur University, Kolkata-32, India) 2 (Department of Electrical Engineering (P.G., 2013 Control System), Jadavpur University, Kolkata-32, India) 3 (Professor, Department of Electrical Engineering, Jadavpur University, Kolkata-32, India) Abstract - In this paper a full order observer has been designed using generalized matrix inverse. The design method resolves the state vector into two unique components, of which one is known and the other is unknown. This method does not assume any structure of the observer and imposes no restriction on the output distribution matrix. Condition of existence of such observer is presented with proof. An illustrative numerical example of two loop missile autopilot is also included with simulation results. Keywords- Das & Ghoshal Observer (DGO), Full Order Observer, Generalized Matrix Inverse, Linear Time Invariant (LTI) Systems, Missile Autopilot. I. INTRODUCTION The problem of state estimation of a linear time invariant system (LTI) from the knowledge of its input and output has been discussed by several authors. In 1964 D.G. Luenberger first introduced the concept and it has been shown in [1] that the state vector of a linear system can be reconstructed from a pre supposed dynamics if the difference between the assumed structure and actual system forms a homogeneous ordinary differential equation. The concept is further developed in [2], where the special topics of identity observer, reduced order observer, linear functional observer, stability properties and dual observers are also discussed. In [3] it has been shown that state vector of an nth order system with m independent outputs can be reconstructed with an observer of order (n-m). S.D.G. Cumming [4] also presented a simple design of stable state observers with reduced dynamics. Such observers with reduced order dynamics result in great reduction in observer complexity. In [5] O’ Reilly proposed a construction method of full order observer which also presupposed the observer structure. In 1981 G. Das and T.K. Ghoshal presented a novel approach to reduced order observer design using generalized matrix inverse [6]. Here the observer structure is not pre assumed and leads to a step by step procedure for observer design. In [6] it has been shown that the DGO is computationally simple as the use of generalized matrix inverse avoids the matrix operations like co- ordinate transformation and matrix partition to design the observer parameter matrices. In [7] it is shown that computation of generalized matrix inverse is not harder than matrix multiplication. In [8] a detailed comparative study has been carried out between reduced order Luenberger observer [1,2] and reduced order Das & Ghoshal observer (DGO) [6]. A method for designing Luenberger full order observer using generalized matrix inverse for both time variant and time invariant linear systems is proposed in [9]. Stefen Hui and Stanislaw H. Zak [10] used projection operator approach (Projection Method Observer or PMO) to estimate the states for the systems with both known and unknown inputs. Here the construction procedure considers the decomposition of the state vector into known and unknown components. Unlike DGO, where the unknown component is the orthogonal projection of the known component, PMO considers the components which are skew symmetric in nature. A detailed and exhaustive comparative study, between the unknown input reduced order DGO and the PMO, has been discussed in [11] and it has been shown that the unknown input DGO is computationally simple compared to PMO in observing the states of a system in presence of unknown inputs.. In [12] reduced order Das & Ghoshal observer had been extended to full-order observer using the principle of generalized matrix inverse. The work presented in [12] has been extended in this article to obtain simpler and computationally less complex design of full order observer using the concept of generalized matrix inverse. Similar to [6] & [12], this construction procedure also does not pre assume the observer structure. The observer dynamics in the proposed design is simpler than that in [12] and imposes no restriction on the output distribution matrix. The following notations will be used here:-  represents the field of real numbers; m x n denotes the dimension of a matrix with m rows and n columns.   denotes the Moore-Penrose generalized inverse of matrix .   is the transpose of  and  represents the identity matrix of appropriate dimension.  represents rank of any matrix  . RESEARCH ARTICLE OPEN ACCESS