Control Trajectories versus Controller Parameters

Two different control-related decision sets are common:

the time-varying signal $\mathbf{u}(t)$ or its sampled values; and
parameters $\mathbf{x}_c$ in a policy $\mathbf{u}=\boldsymbol{\kappa}(\mathbf{y};\mathbf{x}_c)$ .

A direct optimized control signal contrasted with a parameterized controller that maps measurements into control.

Trajectory optimization chooses a signal; controller-parameter optimization chooses a rule that generates the signal from information.

Optimizing trajectories¶

A discretized trajectory may use decisions

\mathbf{u}_1,\mathbf{u}_2,\ldots,\mathbf{u}_N.

(1)

This offers high flexibility, reveals best-case performance, and matches optimal-control theory. However, the result may be specific to one disturbance realization, difficult to implement directly, and does not itself define feedback.

Optimizing controller parameters¶

Choose a realizable structure

\mathbf{u}(t)=\boldsymbol{\kappa}(\mathbf{y}(t);\mathbf{x}_c),

(2)

where $\mathbf{y}(t)$ contains available measurements. Parameters may be PID gains, state-feedback gains, estimator settings, basis weights, or MPC tuning weights.

This representation is directly implementable, explicitly feedback-based, and often lower dimensional. Its limitation is structural: a restricted controller may sacrifice ideal performance, and selecting the controller architecture is itself a design decision.

Why this matters for plant design¶

A plant optimized under unrestricted open-loop control may not remain best when paired with a realistic feedback structure. The control representation changes the attainable closed-loop behavior and can therefore change the optimal physical design.

Activity 6.1: Open-Loop versus Feedback Control Co-Design¶

Activity 6.1: Open-Loop versus Feedback Control Co-Design

Consider the mass-normalized system

\dot{x}_1=x_2, \qquad \dot{x}_2=-kx_1-cx_2+u+d(t).

(3)

The plant variables are

0.5\leq k\leq 8, \qquad 0.1\leq c\leq 3,

(4)

and the actuator-capacity variable is

0.5\leq F_{\max}\leq 5.

(5)

Use the feedback law

u(t)= \operatorname{sat} \left( -K_px_1(t)-K_dx_2(t), F_{\max} \right),

(6)

with

0\leq K_p\leq 10, \qquad 0\leq K_d\leq 8.

(7)

Minimize

\begin{aligned} J=& \frac{1}{N_s} \sum_{s=1}^{N_s} \int_0^8 \left[ x_{1,s}^2 +0.1x_{2,s}^2 +0.02u_s^2 \right]dt \nonumber\\ &+ 0.01k^2 + 0.02c^2 + 0.04F_{\max}^2. \end{aligned}

(8)

Use the scenarios

\mathbf{x}_0^{(1)}= \begin{bmatrix} 1\\ 0 \end{bmatrix}, \qquad \mathbf{x}_0^{(2)}= \begin{bmatrix} 0.5\\ 1 \end{bmatrix}, \qquad \mathbf{x}_0^{(3)}= \begin{bmatrix} -1\\ 0.5 \end{bmatrix},

(9)

with

d_1(t)=0, \qquad d_2(t)=0.4\sin(2t), \qquad d_3(t)=0.3\sin(3t).

(10)

Derive the unsaturated closed-loop dynamics.
Derive the conditions on $k$ , $c$ , $K_p$ , and $K_d$ for asymptotic stability.
Add the passive-safety constraint
$\zeta_{\mathrm{passive}} =\frac{c}{2\sqrt{k}} \geq 0.15.$
(11)
Optimize
$k,\quad c,\quad F_{\max},\quad K_p,\quad K_d$
(12)
simultaneously.
For the optimized plant, solve a separate OLOC problem in GPOPS-II or Dymos for each disturbance scenario.
Compute the relative implementation gap
$G= \frac{J_{\mathrm{feedback}}-J_{\mathrm{OLOC}}} {J_{\mathrm{OLOC}}}.$
(13)
Explain why the plant optimized with unrestricted OLOC may differ from the plant optimized with the parameterized feedback controller.