This article presents a convex inner approximation approach for mixed <inline-formula><tex-math notation="LaTeX">${H}_2$</tex-math></inline-formula>/<inline-formula><tex-math notation="LaTeX">${H}_\infty$</tex-math></inline-formula> control of a flexure-based nanopositioning system. Generally for such positioning systems, the inevitable existence of model mismatch renders it often times difficult-to-achieve satisfying system performance. Additionally, it is essential to also note that the high-order resonances typically presented are prone to be activated if the controller is not designed appropriately, especially in the case when the control input variation arising from the design is unnecessarily drastic. Therefore, to circumvent the above undesirable possibilities, this work aims to improve the tracking performance with a suitable controller design that effectively suppresses the control input variation. Furthermore, despite the existence of model uncertainties, it is shown that it is possible for a subset of stabilizing controller gains to be characterized appropriately via convex inner approximation, which then further facilitates the determination of the controller by means of convex optimization. Rather importantly, this approach provides a performance guarantee with an optimized limiting bound to the <inline-formula><tex-math notation="LaTeX">$H_2$</tex-math></inline-formula>-norm level (which assures optimal behavior for the system), and also concurrently limits the <inline-formula><tex-math notation="LaTeX">$H_\infty$</tex-math></inline-formula>-norm level within a prescribed attenuation level (which satisfies a prescribed robustness measure). Finally, numerical optimization and comparative experiments are carried out for demonstrative purposes.