State-Based Feedback Controllers

Updated 16 December 2025

State-based feedback controllers are methods that compute control actions directly from the system state to ensure stability and performance in various applications.
They employ designs ranging from static and dynamic feedback to advanced SOS programming and data-driven approaches for both linear and nonlinear systems.
Applications span power systems, aerospace, robotics, and decentralized networks, with techniques such as LQR, pole placement, and observer-based methods enhancing robustness.

State-based feedback controllers constitute a foundational methodology in modern control theory, providing mechanisms for closed-loop stabilization, tracking, robustness, safety enforcement, and optimal performance across linear and nonlinear systems. These controllers generate control actions directly as a function of the system state (or augmented state estimate), resulting in either static, dynamic, rational, or data-driven policies that cover a broad class of applications and theoretical frameworks.

1. Fundamental Concepts and Mathematical Structure

A state-based feedback controller computes the control input $u$ as a function of the system state $x$ , often expressed in the static form $u = Kx$ or the nonlinear form $u = \kappa(x)$ . For general nonlinear or uncertain systems, the state may be augmented to include estimates or additional auxiliary variables, yielding dynamic or adaptive feedback architectures.

The canonical linear time-invariant (LTI) form is

$\dot{x} = Ax + Bu, \quad u = Kx,$

with $K$ designed for stability, performance, or safety according to the system goals.

For nonlinear systems, the feedback may be parametric, rational, or constructed by convex sum-of-squares (SOS) programming over Lyapunov certificates, as in rational polynomial state-feedback (Newton et al., 24 Nov 2025).

Decentralized feedback laws further restrict $u_i = K_i x_i$ to only local or partitioned state measurements, often required for large-scale, networked, or distributed systems (Li et al., 6 Sep 2024).

2. State Feedback Design in Linear and Nonlinear Systems

Linear Systems

Pole Placement: The feedback gain $K$ is chosen so that the eigenvalues of $A + BK$ lie in desired positions, achieved via Ackermann’s formula in controllable LTI systems (Vernekar et al., 2020).
Optimal Control (LQR): $K$ is computed to minimize a quadratic cost, leading to the classical Riccati equation solution (Zhang et al., 2023). The optimal $K$ also emerges as the first block of the optimal finite-horizon disturbance-response controller, with the exponential convergence rate in approximation quantified analytically (Zhang et al., 2023).
Robust Invariance: Linear feedback gains can be computed from system data to ensure robust invariance of a polyhedral set despite bounded disturbances, via SDP and LMI relaxations (Mejari et al., 2023).
Constraint Handling: Augmentations via quadratic programming and control barrier functions allow enforcement of state, input, and output constraints, including anti-windup protection in servo architectures (Lavretsky, 24 Nov 2025).
Data-Driven Design: Controllers can be designed directly from trajectory data (PE conditions, Willems’ lemma), offering performance specifications such as tracking, LQR, and robust pole placement without explicit identification of $A, B$ (Lopez et al., 1 Mar 2024).

Nonlinear and Polynomial Systems

Rational Polynomial Feedback: Controllers of the form $u_k(x) = p_k(x)/q_k(x)$ , with $p_k, q_k$ polynomials and $q_k(x) > 0$ , are obtained as solutions to convex SOS programs. The controller and Lyapunov function are co-designed via alternating convex optimizations, with stability regions and performance guarantees directly encoded as SOS constraints (Newton et al., 24 Nov 2025).
Strict-Feedback Systems: Dynamic high-gain scaling and matrix pencil methods allow for state-based stabilizing controllers in triangular nonlinear systems. Online computation of controller parameters is realized via generalized eigenvalues associated with matrix pencils derived from Lyapunov inequalities (Krishnamurthy et al., 2022).
Interconnection and Damping Assignment (IDA-PBC): For two-dimensional systems, partial state feedback using Poincaré’s Lemma enables transformation of the matching PDE into a simpler ODE, significantly streamlining the construction of port-Hamiltonian closed-loop systems (Cisneros et al., 1 Oct 2025).

3. Operational Constraints, Safety, and Robustness

State-based feedback methods have been extended to explicitly handle operational and safety constraints:

Explicit State and Output Constraints: Forward invariance via Nagumo’s theorem and the Comparison Lemma yields control laws ensuring state and input variables remain within prescribed polyhedral boxes. These laws can be synthesized via min-norm quadratic programs resulting in continuous, piecewise-linear feedback (Lavretsky, 24 Nov 2025).
Disturbance Rejection: Extended state observers generate simultaneous estimates of the plant state and unmeasured disturbances, yielding feedback laws incorporating both disturbance compensation and state regulation with rigorous boundedness guarantees (Hu et al., 2018).
Switched Safety Architectures: For uncertain or learned feedback gains, safety can be enforced by switching to a certified fallback controller when the system state exceeds a threshold, guaranteeing bounded long-run cost and safety even for destabilizing learned gains; the resulting closed-loop cost is provably close to optimal under rare fallback events (Lu et al., 2022).
Dissipativity and Input Delay Compensation: Dynamical state feedback controllers using distributed delay kernels are synthesized via Lyapunov-Krasovskii functionals and inner convex approximation of BMIs, extending stability and dissipativity to input delay systems (Feng et al., 2022).

4. Advanced Nonlinear and Infinite-Dimensional Extensions

Nonlinear Output Feedback via Information State: For partially observed nonlinear systems, the information state approach constructs a closed-loop system in terms of stacked past outputs and inputs, transforming the output-feedback problem into a fully observed feedback design. The resulting state-based controller in $\zeta$ -space achieves equivalence in cost and policy with the original partially observed system, enabling the use of iLQR and ARMA identification for complex nonlinear dynamics (Goyal et al., 2021).
Observer-Based Port-Hamiltonian Stabilization: For infinite-dimensional port-Hamiltonian systems, observer-based state feedback via LMI design achieves exponential stabilization, with closed-loop systems forming power-preserving interconnections with strict input/output passivity and zero-state detectability (Toledo et al., 2020).
Temporal Logic and Abstraction-Based Synthesis: State-based output feedback with state-predicate LTL objectives can be approached via abstraction-based synthesis, using grid-based abstraction/refinement, sound predicate mapping, and automata-based reactive synthesis to construct output-feedback controllers guaranteeing temporal-logic specifications (Schmuck et al., 2021).

5. Decentralized, Distributed, and Learning-Based Feedback

Decentralized Control Synthesis: For linear deterministic systems with distributed information, decentralized state-feedback controllers are derived using the matrix maximum principle and a reformulated Riccati equation, with gains obtained via block-structured gradient descent. This framework produces (locally) optimal controllers subject to information constraints (Li et al., 6 Sep 2024).
Learning- and Data-Based State Feedback: Optimization and robust pole-placement for LTI systems can be performed entirely from measured trajectories without model identification, using persistently exciting data and convex programs that guarantee closed-loop constraints and invariance (Lopez et al., 1 Mar 2024).

6. Applications and Performance Analysis

State-based feedback controllers are central to critical applications such as:

Application Domain	Controller Structure	Performance Attributes
Power-electronic conversion (grid-forming)	Full-state feedback	Exact pole placement, robustness
Aerospace/flight systems	Constrained feedback	Safety, anti-windup, MIMO margins
Power system regulation (SMIB, grid)	Pole place/LQR	Robustness over operating points
Large scale/decentralized systems	Decentralized feedback	Scalability, separability
Nonlinear converters, robotics	Rational IDA-PBC/SOS	Global stability, adaptive control

For example, in grid-forming converter systems, full-state feedback compensates for natural coupling, allowing arbitrary eigenvalue placement and robust performance under varying impedance conditions (Chen et al., 2022). In power system SMIB models, both LQR and pole-placement state feedback demonstrate stability across varying load points, with nonlinear feedback-linearizing controllers offering operating-point invariance (Vernekar et al., 2020).

Benchmark studies in nonlinear domains reveal that rational polynomial feedbacks via SOS methods deliver larger regions of attraction, lower cost, and improved robustness compared to classical polynomial-only feedback (Newton et al., 24 Nov 2025). Real-time experimental results in DC-DC converters validate the efficacy of constructive partial state feedback IDA-PBC in fast voltage regulation under highly uncertain load conditions (Cisneros et al., 1 Oct 2025).

7. Outlook and Theoretical Limitations

Despite their versatility, state-based feedback controllers face limitations:

Partial Observability: Constructing state-based laws may require augmentation by observers or information-state mechanisms to handle output-only measurements, as in the information-state approach or observer-based output feedback.
Computational Complexity: Nonlinear, rational, SOS-based, and abstraction-based methods may exhibit substantial computational requirements, particularly in high-dimensional or grid-based settings.
Model Uncertainty and Learning: Data-driven feedback methods mitigate the need for explicit modeling but may require persistency of excitation and sufficient data to guarantee identification of invariance-inducing gains and sets.
Decentralization: Achieving global optimality under decentralized information is generally hard; current methods only yield local minima or require non-convex optimization (Li et al., 6 Sep 2024).
Conservatism, Robustness–Performance Trade-offs: S-procedure relaxations and constraint conservatism may limit the feasible region. Techniques such as robust pole-conditioning and iterative volume maximization address but do not fully eliminate conservatism (Mejari et al., 2023, Lopez et al., 1 Mar 2024).

Ongoing research directions include scalable synthesis for high-dimensional systems, integration with reinforcement learning for safe policy improvement, and unified frameworks for safety-critical constrained and data-driven control in uncertain environments.