Integrated topology and controller optimization of motion systems in the frequency domain

Abstract

In the precision mechatronics industry the limits of current technology are expanded by a very tight integration of the design disciplines involved, such as structural mechanics and control systems technology. In this article a framework is presented which allows the simultaneous and integrated design of a structure and controller. The structure is designed using topology optimization and the objectives and constraints are related to the closed-loop control performance and defined in the frequency domain. For simple examples it is shown that this approach leads to at least as good performance as a sequential approach consisting of eigenfrequency optimization followed by controller tuning, and leads to lighter designs. The framework allows rapid development of prototype designs, which may save a number of the costly design iterations which are currently made in industrial practice. Examples are found in the semiconductor industry, microscopy, micro–electromechanical devices, medical devices and robotics.

Appendices

Appendix A: Some rules for gradients involving complex variables

We consider real-valued functions of complex variables, e.g.:

$$\begin{array}{@{}rcl@{}} f = f(u)\in\mathbb{R},\qquad u=u(\theta)\in\mathbb{C},\qquad \theta\in\mathbb{R}. \end{array} $$

In this case u in turn depends on a real parameter 𝜃. In the context of this article f could be the magnitude of the sensitivity function at a given frequency, the complex variable u could be the harmonic response of the structure, which, in turn, depends on its parameters 𝜃. The real-valued total derivative of f with respect to 𝜃 can be obtained as follows: The function f can always be written as a function of the real and imaginary parts of u=u _R +j u _I and then it is readily seen that (Sarason 2007):

$$\begin{array}{@{}rcl@{}} f = f(u_{{\Re}},u_{{\Im}}) \rightarrow \frac{d f}{d \theta} = \frac{\partial f}{\partial u_{{\Re}}}\frac{d u_{\Re}}{d \theta} + \frac{\partial f}{\partial u_{\Im}}\frac{d u_{\Im}}{d \theta} \end{array} $$

For conciseness, partial derivative operators can be defined according to³:

$$\begin{array}{@{}rcl@{}} \frac{\partial f}{\partial u} &= \frac{\partial f}{\partial u_{{\Re}}} + j\frac{\partial f}{\partial u_{\Im}}, \end{array} $$

(19)

$$\begin{array}{@{}rcl@{}} \frac{\partial u}{\partial \theta} &= \frac{\partial u_{{\Re}}}{\partial \theta} + j\frac{\partial u_{\Im}}{\partial \theta}. \end{array} $$

(20)

With these definitions it is possible to show that:

$$\begin{array}{@{}rcl@{}} \frac{d f}{d \theta} = {\Re}\left\{\frac{\partial f}{\partial \bar{u}} \frac{d u}{d \theta}\right\}, \end{array} $$

in which $\bar {u}$ denotes the complex conjugate of u. This shows that the total derivative is always real-valued, although the partial derivatives involved may be complex-valued.

There is also a vector extension to these rules. When taking gradients of a scalar function f with respect to a vector ${u}\in \mathbb {R}^{n}$ the following notation is adhered to:

$$\begin{array}{@{}rcl@{}} \frac{d f}{d {u}} = \left(\begin{array}{lllll} \frac{d f}{d u_{1}} & {\cdots} & \frac{d f}{d u_{n}}\end{array}\right)^{\top},\qquad \frac{d f}{d {u}^{\top}} = \left(\begin{array}{lllll} \frac{d f}{d u_{1}} & {\cdots} & \frac{d f}{d u_{n}}\end{array}\right). \end{array} $$

For the complex-valued case, let ${u}= {u}(\theta )\in \mathbb {C}^{n}$ be a vector, then the following holds:

$$\begin{array}{@{}rcl@{}} \frac{d f}{d \theta} = \frac{\partial f}{\partial {u}_{{\Re}}^{\top}}\frac{d {u}_{{\Re}}}{d {\theta}} + \frac{\partial f}{\partial {u}_{\Im}^{\top}}\frac{d {{u}}_{\Im}}{d {\theta}} = {\Re}\left\{\frac{\partial f}{\partial {u}^{\star}} \frac{d {{u}}}{d {\theta}}\right\}. \end{array} $$

The superscript ⋆ denotes the complex conjugate transpose; ${u}^{\star }\triangleq \bar { {u}}^{T}$. For a complete overview of the calculus of complex functions see, e.g., (Sarason 2007).

Appendix B: Formulating a Lagrangian for complex-valuedconstraints

The Lagrange multiplier approach for adjoint sensitivity analysis of a function consists in augmenting the function with the equality constraints in the problem (Strang 2007, Section 8.7). For problems such as compliance minimization, subject to a (real-valued) structural compatibility equation of the form K u=f, this constraint is typically expressed using a Lagrange multiplier term of the form:

$$ {\lambda}^{\top}({K}{u}-{f}), $$

which, if the compatibility is satisfied, should be zero for arbitrary nonzero λ.

Now, consider instead the dynamic problem in (1), which can be expressed as K _d u=f, where u is complex-valued. Suppose the residual r of this equation is defined as r=K _d u−f, then the inner product of a complex-valued Lagrange multiplier with this residual yields:

$$\begin{array}{@{}rcl@{}} {\lambda}^{\star} {r} = {\Re}\{{\lambda}\}^{\top}{\Re}\{{r}\}+{\Im}\{{\lambda}\}^{\top}{\Im}\{{r}\} + \\ j({\Re}\{{\lambda}\}^{\top}{\Im}\{{r}\}-{\Im}\{{\lambda}\}^{\top}{\Re}\{{r}\}). \end{array} $$

It follows that if r is to vanish for arbitrary nonzero and complex-valued λ, a necessary and sufficient condition is to require:

$$ {\Re}\left\{{\lambda}^{\star} {r}\right\} = 0. $$

(21)

Note that an equivalent result could be obtained using:

$$ {\Re}\left\{{\lambda}^{\top} {r}\right\} = 0. $$

In this article the former version (21) with the complex conjugate transpose is used. Further note that this condition is equal to:

$$ {\Re}\left\{{\lambda}^{\star} {r}\right\} = \frac{1}{2}\left(\bar{{\lambda}}^{T} {r} + {\lambda}^{T}\bar{{r}}\right) = 0, $$

which is also sometimes found in literature in various forms.