LEGENDRE-TAU APPROXIMATION FOR FUNCTIONAL DIFFERENTIAL EQUATIONS PART II: THE LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM
Original Publication Date: 1899-Dec-30
Included in the Prior Art Database: 2007-Apr-13
Software Patent Institute
Ito, Kazufumi: AUTHOR [+3]
AbstractLEGENDRE-TAU APPROXIMATION FOB FUNCTIONAL DIFFERENTIAL EQUATIONS PART 11: THE LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM Kazufumi Ito
LEGENDRE-TAU APPROXIMATION FOB FUNCTIONAL DIFFERENTIAL EQUATIONS
PART 11: THE LINEAR QUADRATIC OPTIMAL CONTROL PROBLEM
Institute for Computer Applications In Science and Engineering and
University of Vermont
The numerical scheme based on the Legendre-tau approximation is proposed to approximate the feedback solution to the linear quadratic optimal control problem for hereditary differential systems. The convergence property is established using Trotter ideas. The method yields very good approximations at low orders and provides an approximation technique for computing closed- loop eigenvalues of the feedback system. A comparison with existing methods (based on "averaging" and "spline" approximat:ions) is made.
Research was supported by the National Aeronautics and Space Administration under NASA Contracts No. NAS1-17070 and NAS1-17130 while the first author was i n residence at ICASE, NASA Langley Research Center, Hampton, VA 23665.
This paper is the continuation of the study  on the use of Legendre-tau
approximation for functional differential equations (FDE) and concerns the
problem of constructing feedback solutions to linear quadratic regulator
problems for hereditary systems. This problem has received a rather extensive
study and we refer to [I41 , [2 I and  for the summary of the earlier
contributions. Our approach is based upon the pioneering work of Banks -
Burns  who clarified the idea of approximating FDE by systems of finite
dimensional ordinary differential equations and applied it to optimal control
problems; i.e., the convergence of a particular numerical scheme (so-called
'averaging' approximation) is established, using the Trotter-Kato theorem of
linear semigroups. Recently, Gibson  has developed the approximation
theory for the Riccati equations associated with a hereditary system and
applied it to the averaging approximation scheme.
The purposes of this paper are: (i) to apply the basic idea developed in
 to the linear quadratic regulator problem, (ii) to prove convergence of
numerical approximations of the feedback control laws and, (iii) to
demonstrate the feasibility of our numerical schemes.
For the multiple point delay case, the solution to the algebraic Riccati
equation (ARE) has jump discontinuities as shown in . With this
consideration, an extended version of the scheme described in  is developed
for such a case in Section 3. As pointed out in  and will be discussed in
Sections 3 and 5, the tau approximation differs from the standard Galerkin
approximation and because of this, the theory developed in [81 needs to be
modified to prove convergence of approximate solutions to ARE.