TABLE 11.

Double-integrator plant (with transport theorem) with control design based off double-integrator with transport theorem: displays the quantitative comparative data corresponding to the qualitative display in Figures 5A, B.

Method	State error	Rate error	Cost	Runtime
Classical p + V	0.0078728	0.038016	27.0064	3.5535
LQR Optimal PD	−0.0063144	0.57121	75.7706	3.5738
Time-optimal control	0.16359	1.2712	2.7286	3.629
Open loop optimal a	$3.0287\times {10}^{-13}$	$-1.0092\times {10}^{-12}$	7.0286	3.4504
Real-time optimal (RTOC) a	$-9.1882\times {10}^{-6}$	0.019288	7.7942	3.6725
Switched RTOC a	$3.0287\times {10}^{-13}$	$-1.0092\times {10}^{-12}$	7.0286	3.5765

^aUnlike the results of Figure 4 where only linear, time-varying control designs ${\text{u}}_{\text{total}}^{\text{*}}=\widehat{a}t+\widehat{b}$ were used, in Figure 5 nonlinear designs for real-time optimal control ${\text{u}}_{\text{total}}^{\text{*}}=\widehat{a}t+\widehat{b}+{\overline{\omega }}^{\ast }\times \overline{\overline{J}}{\overline{\omega }}^{\ast }$ (with and without switching) were used. Notice the open-loop optimal control is visually indistinguishable from nonlinear, time-varying control designs in the graphic depiction.