Iterative Sequential Design - Control Co-Design

Iterative sequential design alternates plant and controller optimization until changes become small.

A loop alternating between plant optimization and controller optimization until convergence.

Repeated information exchange provides more coordination than a single pass.

An abstract update is

\mathbf{x}_p^{(k+1)}=\arg\min_{\mathbf{x}_p}J(\mathbf{x}_p,\mathbf{x}_c^{(k)}), \qquad \mathbf{x}_c^{(k+1)}=\arg\min_{\mathbf{x}_c}J(\mathbf{x}_p^{(k+1)},\mathbf{x}_c).

(1)

The plant step uses the current controller, and the controller step uses the updated plant.

Advantages¶

Can substantially improve on a single-pass design.
Preserves separate plant and controller subproblems.
Reuses legacy models and tools.
Is easier to implement than full simultaneous CCD.

Limitations¶

Convergence is not guaranteed.
Results can depend on update order.
The process may cycle or converge slowly.
The final point can differ from a fully coordinated optimum.

In strongly coupled problems, each improvement in one discipline changes the best response of the other, making repeated alternation expensive.

Example¶

A wind-turbine team may alternate structural redesign and pitch-controller retuning. Performance can improve gradually, but many costly simulation-based iterations may be required.

Activity 5.2: Convergence of Iterative Sequential Design¶

Activity 5.2: Convergence of Iterative Sequential Design

Consider the quadratic CCD problem

J(\mathbf{p},\mathbf{c}) =\frac{1}{2}\mathbf{p}^TA\mathbf{p} +\mathbf{p}^TB\mathbf{c} +\frac{1}{2}\mathbf{c}^TC\mathbf{c} -\mathbf{a}^T\mathbf{p} -\mathbf{b}^T\mathbf{c},

(2)

where $A$ and $C$ are symmetric positive-definite matrices. An iterative sequential method performs

\begin{aligned} \mathbf{p}^{(k+1)} &=\arg\min_{\mathbf{p}}J\left(\mathbf{p},\mathbf{c}^{(k)}\right),\\ \mathbf{c}^{(k+1)} &=\arg\min_{\mathbf{c}}J\left(\mathbf{p}^{(k+1)},\mathbf{c}\right). \end{aligned}

(3)

Derive the explicit update equations
$\begin{aligned} \mathbf{p}^{(k+1)} &=A^{-1}\left(\mathbf{a}-B\mathbf{c}^{(k)}\right),\\ \mathbf{c}^{(k+1)} &=C^{-1}\left(\mathbf{b}-B^T\mathbf{p}^{(k+1)}\right). \end{aligned}$
(4)
Let $\mathbf{c}^*$ denote the simultaneous optimum. Show that the controller error satisfies
$\mathbf{c}^{(k+1)}-\mathbf{c}^* =M\left(\mathbf{c}^{(k)}-\mathbf{c}^*\right),$
(5)
where
$M=C^{-1}B^TA^{-1}B.$
(6)
Prove that the iterative sequential method converges for every initial controller if
$\rho(M)<1,$
(7)
where $\rho(\cdot)$ denotes spectral radius.
Use
$A= \begin{bmatrix} 4&1\\ 1&3 \end{bmatrix}, \qquad C= \begin{bmatrix} 3&0.5\\ 0.5&2 \end{bmatrix},$
(8)
and
$B= \begin{bmatrix} 1.8&0.4\\ 0.2&1.2 \end{bmatrix}, \qquad \mathbf{a}= \begin{bmatrix} 2\\ 1 \end{bmatrix}, \qquad \mathbf{b}= \begin{bmatrix} 1\\ 2 \end{bmatrix}.$
(9)
Compute $\rho(M)$ and determine whether the iterative sequential method converges.
Starting from
$\mathbf{c}^{(0)}= \begin{bmatrix} 0\\ 0 \end{bmatrix},$
(10)
perform ten alternating iterations and compare the result with the simultaneous optimum.
Explain how the strength of plant–controller coupling is reflected in $\rho(M)$ .