Controller Design for Linear Systems via Controllability Gramian Shaping

This paper presents an optimization-based feedback gain determination by shaping the controllability Gramian, which enables us to design how the input to the system influences its state. First, we define the controllability Gramian for both the stable and unstable systems. The defined controllability Gramian depends on the controller gain, so we formulate the controller design as the problem of shaping the controllability Gramian to achieve the desired relationship between the inputs and states. The proposed method based on an optimization problem evaluates an affine invariant Riemannian metric while preserving the positive-definite symmetric structure of the Gramian. This study describes that the considered controllability Gramian coincides with state covariance in specific settings, and hence, our proposed method can also be utilized for designing the influence from disturbance to the states. Finally, three examples demonstrate the effectiveness of the proposed method.