Stabilizing Output Feedback Controllers

• Stabilizing output feedback controllers are control laws that use only measured outputs to ensure closed-loop system stability despite unmeasured states and noise.
• They deploy methodologies like Lyapunov drift constraints, convex optimization, and observer designs to handle system uncertainties and operational limitations.
• Advanced techniques, including data-driven, quantized, and distributed approaches, guarantee robust stabilization for linear, nonlinear, and hybrid systems.

A stabilizing output feedback controller is a dynamic or static law that utilizes only output (measured) signals to generate control inputs and thereby renders the closed-loop system asymptotically or mean-square stable. In contrast to state feedback, output feedback does not assume access to the full state vector, and must compensate for unmeasured states, often in the presence of system uncertainties, constraints, and measurement noise. The stabilization of general nonlinear, linear, or hybrid systems via output feedback is a central and subtle problem in control theory, with architectures and guarantees that depend crucially on system class, structure, and available measurements.

1. Output Feedback Stabilization: System Classes and Problem Statements

The problem of stabilizing a dynamical system via output feedback arises for both linear time-invariant (LTI) and nonlinear (NL) plants. The canonical LTI form is

$$\begin{aligned} x_{k+1} &= A x_k + B u_k + w_k,\ y_k &= C x_k + v_k, \end{aligned}$$

where $x_k$ is unmeasured, $u_k$ is the control input, and $y_k$ is the measured output. Noise processes $w_k, v_k$ may be present, and constraints such as $u_k \in \mathcal{U}$ (hard bounds) are often imposed (Mishra et al., 2018).

For nonlinear plants, the structure is

$\dot x = f(x,u), \quad y = h(x,u)$

or in a normal form with internal ("zero") and "external" (output-to-integrator-chain) coordinates as in non-minimum phase systems (Boker et al., 2016, Ha et al., 2019). For PDE and hybrid systems, boundary measurements or impulsive events may structure the output channel (Hasan, 2016, Holicki et al., 2019, Sanfelice et al., 2013).

The output-feedback stabilization objective is to construct a controller (static or dynamic, possibly hybrid, possibly constrained) so that for any initial system and controller state, the closed-loop system exhibits asymptotic (or mean-square) stability, often with robustness and constraint satisfaction.

2. Linear and Stochastic Output Feedback: Convex MPC, Drift Conditions, and Mean-Square Stability

For LTI systems with Gaussian noise, incomplete state measurement, and hard control constraints, (Mishra et al., 2018) constructs a stabilizing output feedback stochastic predictive controller. The control law employs a Kalman filter to estimate states and an affine innovation-feedback policy: $$u_k = \bar u_k + \sum_{i=0}^{k} \Theta_{k,i} \varphi(\eta_i)$$ where $\varphi$ is a saturated, bounded odd function. The stabilizing property is achieved by embedding Lyapunov-type drift constraints for the "orthogonal" (marginally stable) modes of $A$: $\E\left[z_j(k+N) - z_j(k)|z(k)\right] \leq -\delta, \text{ for } |z_j(k)| > r$ ensuring mean-square boundedness for any finite input constraint $U_{\max}$. The resulting certifiably stabilizing controller is synthesized recurrently as a convex quadratic program (QP), with stability established via decomposition, drift, and moment arguments.

Key assumptions for this methodology:

• $(A,B)$ stabilizable, $(A,C)$ observable, $A$ Lyapunov stable (no eigenvalues outside unit disk, no Jordan blocks for $|\lambda|=1$), noise processes independent and Gaussian.

3. Quantized and Sampled-Data Output Feedback Controllers: Data-rate and Information Constraints

Quantized output feedback stabilization introduces limitations on the fidelity of output and input signals, due to finite data channels (Wakaiki et al., 2017). A Luenberger observer running at both encoder and controller reconstructs states, with quantizers "zooming" on dynamic centers at each sampling instant. Stabilization is possible provided explicit data-rate lower bounds are satisfied: $N > \frac{M}{1-\rho}$ where $N$ is the number of quantization levels, and $M,\rho$ are parameters from observer dynamics. For systems with joint input and output quantization, coupled bounds on the required number of quantization levels for stability are provided via companion-matrix spectral-radius bounds. Explicit constructions guarantee exponential convergence of quantization errors and closed-loop state.

This quantized architecture is shown to yield dramatically lower bit-rate requirements when high-gain or pseudo-inverse observer designs are used, e.g., for the inverted pendulum benchmark (Wakaiki et al., 2017).

4. Nonlinear Output Feedback Stabilization: Observers, Drift, Sector Bounds, and Hybrid Supervisors

Nonlinear stabilization via output feedback leverages incremental passivity, circle or sector criteria, and composite observer architectures.

• For non-minimum-phase systems, extended high-gain observers (EHGO) recover (output, derivatives), while an EKF tracks internal coordinates. Stability is established using ISS Lyapunov functions and singular perturbation arguments (Boker et al., 2016).
• For MIMO nonlinear plants with uncertain, possibly nonlinear, input gain matrices, disturbance observers with Nyquist-criterion-tuned filters are embedded in the feedback loop to counteract sector-bounded uncertainties:

$\|I - G(z,x,t)\bar G(z,x)^{-1}\| < \mu < 1$

ensuring semi-global practical stability (Ha et al., 2019).

• Hybrid and supervisory control achieves semi-global stabilization (robust basin enlargement) by uniting multiple output-feedback controllers with distinct objectives, orchestrated by hybrid supervisors exploiting norm estimators linked to output-to-state stability properties (Sanfelice et al., 2013).

5. Output Feedback Stabilization via Optimization and System-Level Synthesis

Direct parametrizations and optimization frameworks for output-feedback controllers have advanced both tractability and scalability:

• Linear input-output parametrizations (Youla, SLS) characterize all internally stabilizing output-feedback controllers via affine constraints on closed-loop maps, enabling convex (LP/SDP) synthesis of stabilizing and even optimal (e.g., $\mathcal{H}_2$/$\mathcal{H}_\infty$-optimal) controllers (Furieri et al., 2019, Conger et al., 2021, Galimberti et al., 2024).
• Dynamic output-feedback SLS problems are vectorized and reformulated as discrete-time dynamic programming problems, admitting fast (order $O(n^3 T)$) solution and scaling to large state dimension and FIR horizons (Conger et al., 2021).
• For uncertain, nonlinear, and input-saturated systems, robust QSR-dissipativity and sector constraints are used to formulate stabilizing static output feedback synthesis as LMI hierarchies. Certified domains of attraction are maximized explicitly (Lima et al., 2021).
• Static output feedback with structured constraints can be efficiently "quick-updated" via minimum-destabilizing-real-perturbation bounds to ensure robust stabilization after model changes, using explicit least-squares corrections (Bahavarnia et al., 2023).

6. Data-Driven and Derivative-Free Design of Stabilizing Output Feedback Controllers

Recent data-driven approaches bypass model identification and access only input–output trajectories:

• For continuous-time LTI systems, stabilization can be certified using only filtered I/O data, with derivative-free Lyapunov-based LMIs depending on the known order of the system (Possieri, 20 Jan 2026). Once sufficient persistently-exciting data is collected, a projected data matrix formulation enables convex (SDP) computation of static stabilizing output feedback gains, independent of any explicit state realization or derivative estimation.
• For unknown or partially observed discrete systems, stable output-feedback policy iteration (SPI) algorithms employ compression-amplification mechanisms, reconstruct extended-state regressors, and guarantee closed-loop stability without the need for an initial stabilizing policy (Li et al., 27 Nov 2025).
• In presence of input–output noise, data-driven conditions for stabilizability are encoded in (filter-based) LMI constraints whose feasibility is both necessary and sufficient for closed-loop exponential stability for all consistent realizations with the data. Explicit SNR thresholds are established (Bosso et al., 14 Nov 2025).

7. Advanced Applications: Distributed, Hybrid, and Infinite-Dimensional Systems

• Separation principles using "control templates" for output feedback stabilization of state-affine or bilinear systems are constructed to ensure local observability and semi-global stability under minimal persistent-excitation properties (Sacchelli et al., 2024).
• For distributed systems, parametrizations of all stabilizing nonlinear output-feedback controllers using operator-theoretic SLS yield flexible and robust designs, leveraging neural network parameterizations for high-performance stabilization under open-loop nonlinearities (Galimberti et al., 2024).
• For PDEs, parabolic or impulsive systems, infinite-dimensional backstepping observers, kernel equations, and clock-dependent Lyapunov functionals (enforced via sum-of-squares programming) enable output feedback stabilization with guaranteed decay under output-based, spatially distributed, or aperiodic measurement/actuation (Hasan, 2016, Holicki et al., 2019).

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The stabilization of arbitrary dynamical systems by output feedback remains an overview frontier spanning convex optimization, dissipativity analysis, observer design, and hybrid and data-driven methods. Theoretical guarantees, tractability, and practical implementation depend critically on structural system properties and the information available in the output channel. Modern output feedback controllers synthesize estimation, optimization, and robustness principles to realize certifiably stable closed-loop behavior under broad operational and informational constraints.