I formulate a system with the following state space model:
$$x(k+1) = Ax(k)+Bu(k)+C\epsilon(k),\\
y(k) = x(k).$$
Here, $x$ contains both the original states and the control inputs. Scalar $u$ and vecor $\epsilon$ are two types of system input. Among the outputs within $y$, I especially care about some of them, namely, vector $v=[v_1,v_2]^T$ and vector $w$.
I want to solve the following problem:
\begin{equation}
\min_{\beta,\gamma} \|\Gamma_{\epsilon w}\|_2,\\
s.t. \|\Gamma_{uv_2}/\Gamma_{uv_1}\|_\infty \leq 1, \\
\|\Gamma_{u v}\|_2\leq k,\\
2\beta+\gamma\leq l,
\end{equation}
where $\beta$ and $\gamma$ are two parameters that linearly influence $A$ and $C$ (not the control gain). $k$ and $l$ are two constants. $\Gamma_{\epsilon w}$ denotes the transfer function from $\epsilon$ to $w$ and $\Gamma_{uv_1},\Gamma_{uv_2},\Gamma_{u v}$ are similarly defined. The norms here are $H_2$ or $H_\infty$ norms.
Are there any existing methods for solving such a problem? I've noticed some methods such as LMI are used for $H_\infty$ optimal control. But usually it solves the optimal control gain, which is not my parameters to be optimized.
Best Answer
Sounds like a variant/reparameterization of a mixed H2/Hinf output feedback problem, which is hard. I doubt there is any tractable (i.e. reasonably easily solved) solution to your problem. Here is a pseudo-random link on the topic http://www.math.univ-toulouse.fr/~rondep/Publis/SIAM-H2Hinf.pdf