Static Output Feedback Control

Conside a linear time-invariant state space system \[ \dot{x}(t)=A x(t) +B u(t), \quad y=C x(t) \] with the system matrix $A\in\mathbb{R}^{n\times n}$, the input matrix $B\in\mathbb{R}^{n\times m}$ and the output matrix $C\in\mathbb{R}^{p\times n}$. The system is controlled by a static output feedback controller \[ u(t)=-K y(t) \] with a gain matrix $K\in\mathbb{R}^{m\times p}$.

The problem of static output feedback design is much more complicated than the problem of state feedback design. Using quantifier elimination. the following design probems can be formalized:

  • arbitrary eigenvalue placement
  • special (given) eigenvalue placement
  • interval eigenvalue placement
  • partial eigenvalue placement
  • stabilization
  • real stabilization



  1. Röbenack, K., Voßwinkell, R., Franke, M.: On the eigenvalue placement by static output feedback via quantifier elimination.
    In 2018 26th Mediterranean Conference on Control and Automation (MED), June 2018 (pp. 1-6).
  2. Röbenack, K., Voßwinkel, R., Franke, M., Franke, M.: Stabilization by static output feedback: A quantifier elimination approach.
    In 2018 22nd International Conference on System Theory, Control and Computing (ICSTCC), October 2018 (pp. 715-721).
  3. Röbenack, K., Voßwinkel, R.: Lösung regelungstechnischer Problemstellungen mittels Quantorenelimination.
    at-Automatisierungstechnik, 67(9) 2019, 714-726.
  4. Röbenack, K., Voßwinkel, R.: Eigenvalue placement by quantifier elimination-the static output feedback problem.
    Acta Cybernetica, 24(3) 2020, 409-427.
  5. Röbenack, K., Voßwinkel, R.: On real stable pole placement for structured systems using Sturm and Sturm-Habicht sequences.
    IFAC-PapersOnLine, 53(2) 2020, 4546-4552.
  6. Röbenack, K., Gerbet, D.: Minimum norm partial eigenvalue placement for static output feedback control.
    In: 25th International Conference on System Theory, Control and Computing (ICSTCC), October 2021) (pp. 212-219).
  7. Röbenack, K., Gerbet, D.: Full and Partial Eigenvalue Placement for Minimum Norm Static Output Feedback Control.