Table I.

The NMPC-based DCA that uses gradient projection algorithm.

1	Initialization: j = 0
(a)	Set the convergence tolerance εj.
(b)	Choose initial control trajectory ${\overline{u}}_k^{\left(0\right)}\left(\tau \right)\in {\mathcal{U}}_{\left[t_0,T\right]}$, where τ ∊ [t0, T].
(c)	Integrate the system dynamics forward in time to solve for ${\overline{x}}_k^{\left(0\right)}\left(\tau \right)$ given ${\overline{u}}_k^{\left(0\right)}\left(\tau \right)$ and ${\overline{x}}_k^{\left(0\right)}\left(0\right)=x_k$, where τ ∊ [t0, T].
2	Gradient Step:
(d)	Integrate backward in time to solve for the costates λ(j) (τ) $\dot{\mathrm{\lambda }}\left(\tau \right)={-\frac{\partial H\left(x,\mathrm{\lambda },\mathrm{u}\right)}{\partial x}|}_{x={\overline{x}}_k^{\left(j\right)},u={\overline{u}}_k^{\left(j\right)}},$ where ${\mathrm{\lambda }}_k^{\left(j\right)}\left(T\right)=\frac{\partial V\left({\overline{x}}_k^{\left(j\right)}\left(T\right)\right)}{\partial \overline{x}\left(T\right)}$ is the terminal condition.
(e)	Compute the search direction, $s_k^{\left(j\right)}\left(\tau \right)$, from the Hamiltonian $s_k^{\left(j\right)}\left(\tau \right)={-\frac{\partial H\left(x,\mathrm{\lambda },u\right)}{\partial u}|}_{x={\overline{x}}_k^{\left(j\right)},u={\overline{u}}_k^{\left(j\right)},\mathrm{\lambda }={\mathrm{\lambda }}_k^{\left(j\right)}}.$
(f)	Compute the step size, ${\alpha }_k^{\left(j\right)}$ ${\alpha }_k^{\left(j\right)}=\underset{\alpha >0}{\arg min}J\left(x_k^{\left(j\right)},\psi \left({\overline{u}}_k^{\left(j\right)}+\alpha s_k^{\left(j\right)}\right)\right).$
(g)	Compute the new control trajectory ${\overline{u}}_k^{\left(j+1\right)}\left(\tau \right)=\psi \left({\overline{u}}_k^{\left(j\right)}+{\alpha }_k^{\left(j\right)}{s}_k^{\left(j\right)}\right)$, $\psi \left(u\right)\triangleq \{\begin{array}{ll}{u}^{-},\hfill & u<u^{-}\hfill \\ u,\hfill & u^{-}\le u\le u^{+}\hfill \\ u^{+},\hfill & u^{+}<u\hfill \end{array}.$
(h)	Integrate the system dynamics forward in time (h1) solve for ${\overline{x}}_k^{\left(j+1\right)}\left(\tau \right)$ given ${\overline{u}}_k^{\left(j+1\right)}\left(\tau \right)$, (h2) evaluate the cost function $J\left(x_k^{\left(j+1\right)},{\overline{u}}_k^{\left(j+1\right)}\right)$·
(i)	Check Quit Conditions(i1) quit if $\left|J(x_k^{\left(j+1\right)},{\overline{u}}_k^{\left(j+1\right)})-J(x_k^{\left(j\right)},{\overline{u}}_k^{\left(j\right)})\right|\le {\epsilon }_j$, (i2) quit if j has exceeded the max iteration limit, Nt, (i3) otherwise set j = j + 1 and reiterate gradient step from (a).