Static Output Feedback (SOF) Control

• Static Output Feedback (SOF) involves controlling a continuous-time Linear Time-Invariant (LTI) system using output feedback.

System Equations

• System state is represented by $\dot{x}(t) = Ax(t) + Bu(t)$.
• System output is represented by $y(t) = Cx(t)$.
  • $x(t)$ is the state vector in $\mathbb{R}^n$.
  • $u(t)$ is the input vector in $\mathbb{R}^m$.
  • $y(t)$ is the output vector in $\mathbb{R}^p$.
  • $A$, $B$, and $C$ are constant matrices.
• The static output feedback control law is $u(t) = -Ky(t)$, where $K \in \mathbb{R}^{m \times p}$ is the output feedback gain matrix.
• Resulting closed-loop system: $\dot{x}(t) = (A - BKC)x(t)$.
• The goal of the SOF problem is to find a $K$ matrix so that the closed-loop system is stable, achieved when the eigenvalues of $(A - BKC)$ are in the open left-half plane.
• SOF is a non-convex and NP-hard problem.

SOF Approaches

• Algebraic Riccati Equation (ARE) methods.
• Linear Matrix Inequality (LMI) methods.
• Iterative methods.
• Intelligent optimization algorithms.

Pole Placement

• Pole Placement involves finding a gain matrix $K$ such that the eigenvalues of $(A - BKC)$ match a desired set of closed-loop poles $\lambda_1, \lambda_2,..., \lambda_n$.
• Selecting suitable closed-loop poles is difficult as it's a non-convex problem.

Pole Placement Approaches

• Ackermann's formula.
• Bass-Gura formula.
• LMI-based methods.
• Intelligent optimization algorithms.

Stabilization

• Stabilization aims to find a gain matrix $K$ that ensures the closed-loop system is stable, which means the eigenvalues of $(A - BKC)$ are located in the open left-half plane.
• Stabilization is a non-convex and NP-hard problem.

Stabilization Approaches

• Lyapunov stability theory methods.
• Riccati equation methods.
• LMI-based methods.
• Iterative methods.
• Intelligent optimization algorithms.

$H_2$ Performance

• Involves minimizing the $H_2$ norm of the closed-loop transfer function from disturbance input to controlled output.
• $H_2$ performance is a non-convex problem.

$H_2$ Performance Approaches

• Riccati equation methods.
• LMI-based methods.
• Iterative methods.

$H_\infty$ Performance

• $H_\infty$ Performance involves minimizing the $H_\infty$ norm of the closed-loop transfer function from disturbance input to controlled output.
• $H_\infty$ performance is a non-convex problem.

$H_\infty$ Performance Approaches

• Riccati equation methods.
• LMI-based methods.
• Iterative methods.