Direct method approximation to the state regulator control problem using a Ritz-Trefftz suboptimal control

Author

Bosarge, W. Edwin, Jr. ; Johnson, Olin G.

Author_Institution

IBM Scientific Center, Houston, TX, USA

Volume

Issue

fYear

1970

fDate

12/1/1970 12:00:00 AM

Firstpage

627

Lastpage

631

Abstract

The linear quadratic cost control problem $\\dot{x}(t) = A(t)x(t) + B(t)u(t)$ $x(0) = x_{0}$ with a cost functional $J[u] = frac{1}{2} \\int\\min{0}\\max {T} [\\langle x, Q(t)x\\rangle + \\langle u, R(t)u\\rangle ] dt$ is considered, supposing $S$ is a suitable space of piecewise cubic polynominals on a mesh of norm $h$ on the interval $[0, T]$ . Then a Ritz type algorithm is developed for minimizing $J [\\cdotp]$ over $S$ . The authors have previously discussed [3] certain convergence properties of the algorithm. Here the algorithm is discussed in a form suitable for real-time implementation and additional convergence criteria are presented. In [3] it was shown that the Ritz-Treffiz suboptimal control $\\bar{u}$ converges to the optimal control $u\\ast$ with order $0(h^{3})$ . If $x_{\\bar{u}}$ is the trajectory generated by $\\bar{u}$ , then it is shown that $x_{\\bar{u}}$ approximates the optimal trajectory $x\\ast$ to $0(h^{3})$ . Finally, it is shown that $J[\\bar{u}]$ approximates $J[u\\ast ]$ to order $0(h^{6})$ . The numerical properties of the algorithm, including speed and accuracy comparisons with the conventional numerical approach, are presented in a forthcoming paper.

Keywords

Linear systems, time-varying continuous-time; Optimal regulators; Suboptimal control; Convergence; Cost function; Distributed computing; Fuzzy control; Linear systems; Minimization methods; Optimal control; Polynomials; Regulators; Size control;

fLanguage

English

Journal_Title

Automatic Control, IEEE Transactions on

Publisher

ieee

ISSN

0018-9286

Type

jour

DOI

10.1109/TAC.1970.1099594

Filename

1099594

Link To Document

https://search.isc.ac/dl/search/defaultta.aspx?DTC=49&DC=804758