An SQP-type Solution Method for Constrained Discrete-Time Optimal Control Problems

Abstract

The considered nonlinear, constrained discrete-time optimal control problem is stated as follows:

$$ J = F({x^N}) + \sum\limits_{k = 0}^{N -1} {f_0^k} ({x^k},{u^k}) \to Min $$

subject to the state equation:

$$ {x^{k + 1}} = {f^k}({x^k},{u^k}),k = 0, \ldots ,N -1, $$

(1)

and inequality constraints:

$$ \begin{gathered} {{c}^{k}}({{x}^{k}},{{u}^{k}}) \leqslant 0,\quad k = 0, \ldots ,N - 1,{\kern 1pt} \hfill \\ {\kern 1pt} {{c}^{N}}({{x}^{N}}) \leqslant 0, \hfill \\ {{f}^{k}}:{{{\text{R}}}^{n}} \times {{{\text{R}}}^{m}} \to {{{\text{R}}}^{n}},{{c}^{k}}:{{{\text{R}}}^{n}} \times {{{\text{R}}}^{m}} \to {{{\text{R}}}^{{{{r}^{k}}}}} \hfill \\ \end{gathered} $$

with sufficiently smooth functions F, ƒ ^k₀ , ƒ ^k, c ^k. The constraints include fixed initial or final states as well as bounds for state and control variables or more general constraints.