Space splitting convexification: a local solution method for nonconvex optimal control problems

Figures & data

Figure 1. Transformation of OP; Dotted and solid arrows represent approximated and exact transformations, respectively.

Figure 2. Space splitting. (a) Splitting of decision variable w into $w_{\mathrm{l}\mathrm{o}\mathrm{w}}$ and $w_{\mathrm{u}\mathrm{p}}$. (b) Absolute value constraint (9b(9b) $\begin{array}{rl}{g}_{\text{abs}}& =\underset{w_{\mathrm{\Delta }}}{\underbrace{w_{\text{up}}-w_{\text{low}}}}-|\underset{\tilde{w}}{\underbrace{w-w_{\text{tr}}}}|=0.\end{array}$(9b) ) and its relaxed linearisation according to (13a(13a) $\begin{array}{rl}& w_{-2\$\mathrm{\Delta }\$}-{\sigma }_w\tilde{w}=s_w\end{array}$(13a) ). (c) Projection after optimisation with initially wrong sign; I: initial guess; O: optimum; P: projection of optimum.

Figure 3. Approximation of nonlinearity $f_{\mathrm{n}\mathrm{l}}$ with three-/five-segmented piecewise linear polynom $p_{\mathrm{p}\mathrm{w}\mathrm{l},3}/p_{\mathrm{p}\mathrm{w}\mathrm{l},5}$; Approximation error can be reduced with an increasing number of splitting segments.

Figure 4. Space splitting convexification of zonally convex sets. (a) ZCS with a single transition point. (b) Shifted, rotated ZCS with a single transition point. (c) ZCS with shared line at transition.

Figure 5. Space splitting convexification of piecewise linear equalities. (a) Nonlinearity with two linear segments. (b) Nonlinearity with three linear segments. (c) Splitting of state variable x for three segments.

Figure 6. Model characteristics for hanging SMO with semi-active damper and nonlinear spring. (a) Hanging SMO model. (b) ZCS for semi-active damper force. (c) Piecewise linear spring force characteristic.

Table 1. Parameters for SMO application.

Symbol	Value	Unit	Description
g	9.81	m/${\mathrm{s}}^2$	gravitational acceleration
m	5.00	kg	mass
$c_{\mathrm{u}\mathrm{p}}$	10.00	N/m	overload spring stiffness for rebound
$c_{\mathrm{m}\mathrm{i}\mathrm{d}}$	3.00	N/m	main spring stiffness
$c_{\mathrm{l}\mathrm{o}\mathrm{w}}$	5.00	N/m	overload spring stiffness for compression
${\overline{x}}_{\mathrm{t}\mathrm{r}}$	5.00	m	transition point for rebound overload
${\underset{\_}{x}}_{\mathrm{t}\mathrm{r}}$	$-5.00$	m	transition point for compression overload
$x_{\mathrm{s}\mathrm{s},2}$	16.35	m	steady-state position for 2-segment spring
$x_{\mathrm{s}\mathrm{s},3}$	8.405	m	steady-state position for 3-segment spring
$\overline{x}$	100.00	m	upper position bound
$\underset{\_}{x}$	$-100.00$	m	lower position bound
$\overline{d}$	20.00	Ns/m	upper bound on damper coefficient
$\underset{\_}{d}$	0.50	Ns/m	lower bound on damper coefficient
$\overline{v}$	100.00	m/s	upper speed bound
$\underset{\_}{v}$	$-100.00$	m/s	lower speed bound
${\overline{F}}_d$	400.00	N	upper bound on damper force
${\underset{\_}{F}}_d$	$-400.00$	N	lower bound on damper force
${ϵ}_g$	${10}^{-6}$	–	violation tolerance for SSC penalty term
$\underset{\_}{\tau }/\overline{\tau }$	$1.00/{10}^4$	–	initial/final value for penalty parameter

Table 2. Initial values of robustness analysis.

	${\mathbf{\text{x}}}_{\mathrm{i}\mathrm{v},1}$	${\mathbf{\text{x}}}_{\mathrm{i}\mathrm{v},2}$	${\mathbf{\text{x}}}_{\mathrm{i}\mathrm{v},3}$	${\mathbf{\text{x}}}_{\mathrm{i}\mathrm{v},4}$	${\mathbf{\text{x}}}_{\mathrm{i}\mathrm{v},5}$	${\mathbf{\text{x}}}_{\mathrm{i}\mathrm{v},6}$	${\mathbf{\text{x}}}_{\mathrm{i}\mathrm{v},7}$	${\mathbf{\text{x}}}_{\mathrm{i}\mathrm{v},8}$	${\mathbf{\text{x}}}_{\mathrm{i}\mathrm{v},9}$	${\mathbf{\text{x}}}_{\mathrm{i}\mathrm{v},10}$
x	−24.5	−15.1	−51.4	27.9	59.4	12.3	56.1	15.4	−73.7	−56.9
v	−30.0	−44.9	−15.4	39.3	18.7	64.3	−15.5	−32.7	16.8	73.1

Table 3. Optimisation results of robustness analysis.

	Spring		Method		Results
$n_{\mathrm{s}\mathrm{e}\mathrm{g}}[-]$	LIN	NL	NLP	SSC	${\overline{J}}_{\mathrm{s}\mathrm{s}}\left[{\mathrm{m}}^2\mathrm{s}\right]$	$\overline{{\overline{e}}_{\mathrm{s}\mathrm{s}}}\left[\mathrm{m}\right]$	$\overline{e_{F_d}}\left[\mathrm{N}\right]$	$\overline{q}[-]$	$t_{\mathrm{o}\mathrm{p}\mathrm{t},\mathrm{\Sigma }}\left[\mathrm{s}\right]$	$t_{\mathrm{\Sigma }}\left[\mathrm{s}\right]$
10	✓		✓		815.26	7.64	–	1.00	0.074	0.074
10	✓			✓	818.92	7.50	3.51	2.00	0.005 (6.45%)	0.021 (27.77%)
10		✓	✓		597.24	7.11	–	1.00	0.075	0.075
10		✓		✓	601.84	6.96	2.72	2.40	0.008 (10.90%)	0.025 (33.67%)
50	✓		✓		2135.09	6.67	–	1.00	0.125	0.125
50	✓			✓	2139.30	6.55	2.04	2.00	0.016 (12.87%)	0.036 (28.51%)
50		✓	✓		1842.48	6.16	–	1.00	0.127	0.127
50		✓		✓	1849.97	6.03	2.28	2.60	0.036 (28.46%)	0.059 (46.34%)
100	✓		✓		2333.84	6.52	–	1.00	0.229	0.229
100	✓			✓	2338.26	6.41	1.94	2.00	0.026 (11.18%)	0.050 (21.77%)
100		✓	✓		2043.91	6.02	–	1.00	0.252	0.252
100		✓		✓	2051.33	5.89	2.20	2.70	0.064 (25.44%)	0.095 (37.53%)
250	✓		✓		2454.05	6.43	–	1.00	0.587	0.587
250	✓			✓	2458.40	6.32	1.94	2.00	0.068 (11.64%)	0.105 (18.20%)
250		✓	✓		2167.57	5.93	–	1.00	0.695	0.695
250		✓		✓	2174.81	5.80	2.22	2.70	0.183 (26.30%)	0.235 (33.83%)
500	✓		✓		2494.96	6.40	–	1.00	1.080	1.080
500	✓			✓	2499.36	6.29	1.97	2.10	0.173 (16.03%)	0.235 (21.72%)
500		✓	✓		2209.03	5.91	–	1.00	1.373	1.373
500		✓		✓	2216.23	5.77	2.22	2.80	0.460 (33.50%)	0.555 (40.39%)

$n_{\mathrm{s}\mathrm{e}\mathrm{g}}$: # of collocation segments; LIN/NL: linear/nonlinear spring; NLP: IPOPT solving NLP problem (51(51a) $\begin{array}{rl}\underset{\mathbf{\text{u}},\mathbf{\text{X}}}{min}& J_{\mathrm{s}\mathrm{s}}\end{array}$(51a) ); SSC: SSC Algorithm 1 with QP problem (55(55a) $\begin{array}{rl}\underset{\mathbf{\text{P}}}{min}& J_{\mathrm{s}\mathrm{s}}+\tau \sum _{k\in K}{l}_{\mathrm{a}\mathrm{b}\mathrm{s},x,k}+l_{\mathrm{a}\mathrm{b}\mathrm{s},v,k}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\end{array}$(55a) ); ${\overline{J}}_{\mathrm{s}\mathrm{s}}$/$\overline{{\overline{e}}_{\mathrm{s}\mathrm{s}}}$/$\overline{e_{F_d}}$: main objective and error metrics (59(59) $\begin{array}{r}{\overline{J}}_{\mathrm{s}\mathrm{s}}\sum _{i=1}^{10}{J}_{\mathrm{s}\mathrm{s},i},\overline{{\overline{e}}_{F_d}}\sum _{i=1}^{10}{\overline{e}}_{F_d,i}\mathrm{a}\mathrm{n}\mathrm{d}\overline{{\overline{e}}_{\mathrm{s}\mathrm{s}}}\sum _{i=1}^{10}{\overline{e}}_{\mathrm{s}\mathrm{s},i}.\end{array}$(59) )(average for 10 initial value conditions); $\overline{q}$: average # of superordinate iterations; $t_{\mathrm{o}\mathrm{p}\mathrm{t},\mathrm{\Sigma }}$: cumulative time spent in NLP/QP solver; $t_{\mathrm{\Sigma }}$: total computation time.

Figure 7. Solution of OPs with initial value ${\mathbf{\text{x}}}_{\mathrm{i}\mathrm{v},1}$ and $n_{\mathrm{s}\mathrm{e}\mathrm{g}}=250$ collocation segments. (a) Piecewise linear spring force. (b) ZCS for damper force. (c) State and damper force trajectories.

Figure 8. Optimisation results for SMO application with three-segmented nonlinear spring for $k_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}=0$. (a) Piecewise linear spring force with three segments. (b) ZCS for damper force.

Table 4. Analysis of sensitivity regarding initial guess.

	method		results
$k_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}[-]$	NLP	SSC	$J_{\mathrm{s}\mathrm{s}}\left[{\mathrm{m}}^2\mathrm{s}\right]$	${\overline{e}}_{\mathrm{s}\mathrm{s}}\left[\mathrm{m}\right]$	${\overline{e}}_{F_d}\left[\mathrm{N}\right]$	$q[-]$	$t_{\mathrm{o}\mathrm{p}\mathrm{t},\mathrm{\Sigma }}\left[\mathrm{s}\right]$	$t_{\mathrm{\Sigma }}\left[\mathrm{s}\right]$
0.0	✓		1603.95	5.47	–	1	4.589	4.589
0.0		✓	2118.49	9.91	46.63	7	0.493 (10.74%)	0.798 (17.38%)
0.5	✓		1603.95	5.47	–	1	6.347	6.347
0.5		✓	1635.20	5.93	14.43	19	1.450 (22.85%)	2.136 (33.65%)
0.75	✓		1603.95	5.47	–	1	0.557	0.557
0.75		✓	1612.62	5.67	15.77	2	0.179 (32.14%)	0.324 (58.26%)
0.95	✓		1603.95	5.47	–	1	0.473	0.473
0.95		✓	1603.95	5.47	0.01	1	0.075 (15.86%)	0.199 (42.02%)
1.0	✓		1603.95	5.47	–	1	0.417	0.417
1.0		✓	1603.95	5.47	0.01	1	0.072 (17.28%)	0.191 (45.79%)
1.05	✓		1603.95	5.47	–	1	0.427	0.427
1.05		✓	1603.96	5.47	0.20	1	0.084 (19.67%)	0.202 (47.37%)
10.0	✓		1603.95	5.47	–	1	0.981	0.981
10.0		✓	1610.74	5.72	6.71	2	0.149 (15.19%)	0.306 (31.22%)

$k_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}$: perturbation parameter for initialisation error; NLP: IPOPT solving NLP problem; SSC: SSC Algorithm 1 with QP problem (64a(64a) $\begin{array}{rl}& \underset{\mathbf{\text{P}}}{min}{J}_{\mathrm{s}\mathrm{s}}+\tau \sum _{k\in K}{l}_{\mathrm{a}\mathrm{b}\mathrm{s},x_{\mathrm{m}\mathrm{u}},k}+l_{\mathrm{a}\mathrm{b}\mathrm{s},x_{\mathrm{l}\mathrm{m}},k}+l_{\mathrm{a}\mathrm{b}\mathrm{s},v,k}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h}\end{array}$(64a) ); $J_{\mathrm{s}\mathrm{s}}$: main objective (48(48) $\begin{array}{r}{J}_{\mathrm{s}\mathrm{s}}\sum _{i=0}^{n_{\mathrm{s}\mathrm{e}\mathrm{g}}-1}\frac{{\mathrm{\Delta }}_i}{6}\left(e_{\mathrm{s}\mathrm{s},i}^2+4e_{\mathrm{s}\mathrm{s},i+\frac{1}{2}}^2+e_{\mathrm{s}\mathrm{s},i+1}^2\right).\end{array}$(48) ); ${\overline{e}}_{\mathrm{s}\mathrm{s}}$/${\overline{e}}_{F_d}$: error metrics (58(58) $\begin{array}{r}{\overline{e}}_{\mathrm{s}\mathrm{s}}\frac{1}{n_{\mathrm{p}\mathrm{t}\mathrm{s}}}\sum _{k\in K}|x_k-x_{\mathrm{s}\mathrm{s},k}|\end{array}$(58) ) and (57(57) $\begin{array}{r}{\overline{e}}_{F_d}\frac{1}{n_{\mathrm{p}\mathrm{t}\mathrm{s}}}\sum _{k\in K}\left|F_{d,k}^{\mathrm{N}\mathrm{L}\mathrm{P}}-F_{d,k}^{\mathrm{S}\mathrm{S}\mathrm{C}}\right|\end{array}$(57) ); q: # of superordinate iterations; $t_{\mathrm{o}\mathrm{p}\mathrm{t},\mathrm{\Sigma }}$: cumulative time spent in NLP/QP solver; $t_{\mathrm{\Sigma }}$: total computation time.

Figure 9. SSC optimisation results for SMO application with fully linear spring and rotated damper set. (a) Rotated ZCS for damper force. (b) Position trajectory; $x^{\mathrm{S}\mathrm{S}\mathrm{C}}$: original damper force set; $x^{\mathrm{S}\mathrm{S}\mathrm{C},\phi }$: rotated damper force set.