Open-Loop Optimal Control - Control Co-Design

A control trajectory is a signal planned over time. A controller is a rule mapping available information to actions during operation. Disturbance, noise, and model uncertainty make exact future behavior unknowable, so most practical systems ultimately require feedback.

Basic formulation¶

Open-loop optimal control (OLOC) chooses $\mathbf{u}(t)$ over $t\in[t_0,t_f]$ using a model, initial condition, and assumed future information:

\begin{aligned} \underset{\mathbf{u}(\cdot),\mathbf{x}(\cdot)}{\text{minimize}}\quad &J=\Phi(\mathbf{x}(t_f))+\int_{t_0}^{t_f}L(\mathbf{x}(t),\mathbf{u}(t),t)\,dt\\ \text{subject to}\quad &\dot{\mathbf{x}}(t)=\mathbf{f}(\mathbf{x}(t),\mathbf{u}(t),t),\\ &\mathbf{x}(t_0)=\mathbf{x}_0,\\ &\mathbf{c}(\mathbf{x}(t),\mathbf{u}(t),t)\leq\mathbf{0},\\ &\mathbf{b}(\mathbf{x}(t_f))\leq\mathbf{0}. \end{aligned}

(1)

A model, initial state, and assumed future information producing one optimized control trajectory that is applied without feedback.

The open-loop trajectory is planned first and then applied as planned.

OLOC can be reasonable when the horizon is short, uncertainty is small, the result is a benchmark, or the trajectory is repeatedly recomputed as in MPC.

Role in CCD¶

OLOC can answer:

What is the best achievable performance for a candidate plant under an assumed scenario?
How much energy can ideally be extracted or saved?
What qualitative features should control action have?
Which plant properties make the control task easier?

For a wave-energy device, optimizing PTO force against a fixed recorded wave provides an upper-bound energy benchmark. It does not automatically produce a realizable controller, but it exposes valuable plant and control structure.