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

The linear quadratic cost control problemẋ(t) = A(t)x(t) + B(t)u(t)x(0) = x₀with a cost functionalJ[u] = 1/2 ∫min{0}max{T} [, Q(t)x + , R(t)u] dtis considered, supposingSis a suitable space of piecewise cubic polynominals on a mesh of normhon the interval[0, T]. Then a Ritz type algorithm is developed for minimizingJ []overS. 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 controlūconverges to the optimal controlu order0(h³). Ifxūis the trajectory generated byū, then it is shown thatxūapproximates the optimal trajectoryx0(h³). Finally, it is shown thatJ[ū]approximatesJ[u]to order0(h⁶). The numerical properties of the algorithm, including speed and accuracy comparisons with the conventional numerical approach, are presented in a forthcoming paper.