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

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} [\langlex, Q(t)x\rangle + \langleu, 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.