6.1 What Open-Loop Optimal Control Means

Core idea¶

What Open-Loop Optimal Control Means must be treated as a system-level decision rather than an isolated technique. For an active suspension with complete road preview and a wave-energy converter with known future waves, state what is fixed, what is optimized, what information is available, and what equations define feasibility.

The relevant quantities are $p$ , $u(t)$ , $x(t)$ , and known $d(t)$ . The chapter-level formulation is

\min_{p,u,x}\Phi+\int L\,dt\;\mathrm{s.t.}\;\dot x=f(t,x,u,p,d).

For this section, trace how the choice changes plant variables, the active constraints, and the implementable engineering design. A method is useful only when its assumptions are explicit and its result answers the same system question as the baseline.

Engineering interpretation¶

Ask three questions:

Which physical, informational, computational, or economic resource changed?
Which objective component or active constraint made the change valuable?
Does the conclusion survive model, disturbance, initialization, uncertainty, and implementation checks?

A practical action is to state information assumption. Record units and assumptions before optimization, report component objectives and margins afterward, and verify the result using an independent calculation or higher-fidelity model.

Activity 6.1: quantitative design audit¶

Activity 6.1: quantitative design audit

A 1.2 kg positioning carriage must move from $x_0=[q_0,v_0]^T=[0,0]^T$ to $x_f=[1,0]^T$ in $T=2$ s. Divide the maneuver into $N=8$ intervals of length $h=0.25$ s and assume the commanded force is constant within each interval. The exact discrete-time model is

x_{k+1}= \begin{bmatrix}1&h\\0&1\end{bmatrix}x_k+ \begin{bmatrix}h^2/(2m)\\h/m\end{bmatrix}(u_k+d_k), \qquad m=1.2~\text{kg}.

A preview sensor supplies the nominal disturbance sequence, in newtons,

d^{\mathrm{pred}}=[0,\;0.30,\;0.60,\;0.60,\;0.30,\;0,\;-0.20,\;-0.20].

Choose $u=[u_0,\ldots,u_7]^T$ to minimize

J=100(q_N-1)^2+20v_N^2+0.02h\sum_{k=0}^{N-1}u_k^2,

subject to $|u_k|\leq3$ N and $|v_k|\leq1.2$ m/s. Use MATLAB fmincon, quadprog, or an equivalent optimizer.

Compute the preview-optimal sequence $u^*$ and report $q_N$ , $v_N$ , $J$ , $\max|u_k|$ , $\max|v_k|$ , and both constraint margins.
Repeat the optimization while assuming $d_k=0$ . Apply this no-preview sequence to the predicted disturbance above. Quantify the change in terminal position error and objective value.
Test implementability without re-optimizing: the actual disturbance is delayed one interval and is 20% larger, $d^{\mathrm{val}}=1.2[0,\;0,\;0.30,\;0.60,\;0.60,\;0.30,\;0,\;-0.20]$ . Apply the original $u^*$ and recompute every metric and constraint margin.
Calculate the percentage of the nominal preview benefit that survives validation, using reduction in $|q_N-1|$ as the benefit metric.
State precisely what future information must be available before $t=0$ to implement $u^*$ . Explain, using your numerical results, why this sequence is an open-loop performance benchmark rather than a disturbance-rejecting feedback controller.