Static Output Feedback Control in Linear Systems

Static output feedback control addresses the design of fixed gain laws that use measured system outputs to generate control actions for linear time-invariant systems. Unlike full state-feedback, where all internal states are assumed available, static output feedback relies solely on a constant gain matrix acting on a subset of system outputs. This approach offers reduced implementation complexity, lower sensor and computational requirements, and greater suitability for decentralised or embedded applications. However, the design problem is inherently non-convex, since stability and performance objectives often lead to bilinear matrix inequalities or non-convex constraints on the gain matrix. Over the past decade, significant efforts have been devoted to convex relaxations using linear matrix inequalities, iterative projection methods and randomized optimisation techniques, all seeking to balance robustness, disturbance attenuation and closed-loop performance. Applications span from aerospace and power electronics to large-scale networked infrastructure, where simplicity of the control law and guaranteed stability are critical.

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Recent advances have introduced necessary and sufficient conditions for output-feedback stabilisability within the linear quadratic regulator framework, alongside iterative Newton-type schemes that converge from a stabilising state-feedback solution to a static output-feedback gain. These methods address high-dimensional systems by reducing the design to successive Lyapunov equations, thereby improving computational tractability and ensuring closed-loop optimality under a quadratic cost.

Another line of work has applied randomized global optimisation to the static output feedback LQR problem, employing the Ray-Shooting Method to navigate highly non-convex synthesis regions. This probabilistic algorithm offers convergence guarantees in probability and demonstrates effective controller synthesis for continuous-time systems, even when classical convexification fails.

Foundational contributions in robust H∞ control have proposed dilated linear matrix inequality conditions for static output feedback. By introducing auxiliary slack variables, these formulations reduce conservatism and extend naturally to polytopic uncertainty via parameter-dependent Lyapunov functions. Numerical studies confirm that the dilated conditions encompass earlier standard LMI approaches, delivering enhanced disturbance attenuation and broader admissible uncertainty bounds.

Static Output Feedback Control in Linear Systems publication trend

The graph below shows the total number of articles in static output feedback control in linear systems across all publications each year (not limited to Nature Index journals).

Technical terms

Static output feedback: A control strategy in which the system output is fed back through a constant gain to compute the control input.

Linear time-invariant (LTI) system: A dynamic system whose behaviour is governed by constant-coefficient linear differential or difference equations.

Linear matrix inequality (LMI): A convex constraint of the form A₀ + x₁A₁ + … + xₙAₙ ≥ 0 used for control design via semidefinite programming.

Linear Quadratic Regulator (LQR): An optimal control problem that minimises a quadratic cost on states and inputs for an LTI system.

H∞ norm: The worst-case gain from an exogenous disturbance to the controlled output, capturing robust performance requirements.

References

1. Novel Results on Output-Feedback LQR Design. IEEE Transactions on Automatic Control (2022).
2. On Application of the Ray-Shooting Method for LQR via Static-Output-Feedback. Algorithms (2018).
3. Sufficient Dilated LMI Conditions for H∞ Static Output Feedback Robust Stabilization of Linear Continuous‐Time Systems. Journal of Applied Mathematics (2012).