E E E TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 39, NO. 4, APRIL 1994 793 zyx Technical Notes and Correspondence A New Orthogonal Series Approach to State Space Analysis of Bilinear Systems P. N. Paraskevopoulos, A. S. Tsirikos, and K. G. Arvanitis Abstruct-In this note, a new orthogonal series approach to state- space analysis of bilinear time-invariant systems is presented. The present approach involves only multiplications of matrices of small dimensions. Known techniques involve the solution of an algebraic system zyxwvutsr with a very large number of equations. I. INTRODUCTION Bilinear systems may be considered zyxwvutsr as a specialization of nonlinear systems under the assumption of linearity in the control or respec- tively in the state but not jointly. This kind of systems has been extensively investigated including studies of structural properties, identification and control problems zyxwvutsrqp [ 11-[6]. This note presents a new orthogonal series approach for the solution of the state-space equation of bilinear time-invariant systems. This approach leads to a simple expression for the coefficient matrix X for the state vector. This expression involves products of matrices of small dimensions. Known results involve the solution of an algebraic system with a very large number of equations [7]-[13]. 11. PRELIMINARIES Consider the bilinear time-invariant system described in state space zyxwvuts z(t = 0) = z(0) (2.1) where z(t) E E", zyxwvuts ~(t) E R" and A, zyxwvutsrqp B and C are constant matrices of appropriate dimensions. The term Nz(t)u(t) is a bilinear form in the variables z(t) and ~(t), which is defined as [3] by the following equation i(t) = Az(t) + Nz(t)u(t) + Bu(t), m NZ(t)U(t) = CN,z(t)u,(t) (2.2) 1=1 where N, E R"'" and ut(t) is the ith component of u(t). given by [3] At The solution of (2.1) for z(t) in terms of a series expansion, is z(t) =e z(0) . N[. . . [&-l-zi )Bu(z;)].**]x(~~)~z; -..dxl. (2.3) Manuscript received April 20, 1990; revised July 3, 1992 and December 21, 1992. This work was supported in part by the Greek State Scholarship Foundation (IKY). The authors are with the National Technical University of Athens, Di- vision of Computer Science, Department of Electrical Engineering, 15773 Zographou, Athens, Greece. JEEE Log Number 9212842. Relation (2.3) may be written in the following equivalent form z(t) = eAtz(0) + lte~(t-zl)Bu(sl) dzl . zyxwvu N[. . . [eA(Zz-l-"t) N[ZL(~Z)l~(~Z)l~~ .I .~(zi)dx, ...&I. (2.7) The first term ZL(~) is the solution for the state vector z(t), for a linear time-invariant system described by the equation z(t) = Az(t) + Bu(t). The second term zp,(t) is due to the bilinear term Nz(t)u(t) appearing in (2.1). III. STATE-SPACE ANALYSIS The term ZL (t) in (2.5) may be approximated via orthogonal series as follows [14] and [151 z~(t) 21 Gof,(t), Go E R"" (3.1) where f,(t) = [fo(t), fl(t), . . . , fr-l(t)lT is the orthogonal basis vector and Go = [z(O) i Az(0) ! ... i A"'z(O)] [ e ' eT Pr ] eT Pt-' ':] (3.2) UP! 0018-9286/94$04.00 zyxwvuts 0 1994 IEEE -~ ~ _ _ _ -