Sequential-Linearization Feedback Optimization

• Sequential-Linearization Feedback Optimization is a method that drives nonlinear systems to optimal steady states by interleaving real-time measurements with locally updated Jacobian approximations.
• It employs a sequential linearization mechanism that refreshes sensitivity estimates at the current operating point, addressing biases inherent in fixed-point approximations.
• The approach has proven effective in high-dimensional applications like wind farm control, balancing computational cost with improved performance and convergence guarantees.

Searching arXiv for the directly relevant SLFO paper and adjacent feedback-optimization/feedback-linearization references. Sequential-Linearization Feedback Optimization (SLFO) is a feedback-optimization paradigm for driving a nonlinear dynamical system to an optimal steady state by interleaving real-time measurements with projected first-order updates that use a locally linearized approximation of the plant’s steady-state sensitivity. In the formulation introduced for nonlinear discrete-time systems, the plant output is measured online, the steady-state input-output Jacobian is approximated from a linearization at the current operating point, and the linearization point is updated sequentially along the optimization trajectory rather than fixed a priori (Huang et al., 20 Jul 2025). The defining motivation is that standard feedback optimization requires accurate first-order information of the steady-state input-output mapping, while a fixed nominal linearization can become badly biased when the operating point moves through a strongly nonlinear region (Huang et al., 20 Jul 2025).

1. Definition and mathematical setting

The canonical SLFO setting uses the nonlinear discrete-time plant

$$\begin{cases} x^{+} = f(x,u) + w_1\ y = g(x,u) + w_2 \end{cases}$$

with state $x\in\mathbb{R}^n$, input $u\in\mathcal U\subseteq \mathbb{R}^p$, measured output $y\in\mathbb{R}^m$, and unknown disturbance $w=\mathrm{col}\{w_1,w_2\}$ (Huang et al., 20 Jul 2025). Under invertibility of $I-\nabla_x f(x,u)$, the steady-state state map $\phi$ is defined implicitly by

$\phi(u,w_1)=f(\phi(u,w_1),u)+w_1,$

and the corresponding steady-state input-output map is

$y_{\text{ss}} = g(\phi(u,w_1),u)+w_2 =: h(u,w).$

The steady-state optimization problem is

$$\begin{cases} \min\limits_{\bar u,\bar y} J(\bar u,\bar y)\ \text{s.t.}\quad \bar y = h(\bar u,w),\quad \bar u\in\mathcal U. \end{cases}$$

A central quantity is the reduced objective $x\in\mathbb{R}^n$0, whose gradient is

$x\in\mathbb{R}^n$1

This shows that even when the unknown steady-state output $x\in\mathbb{R}^n$2 is replaced by real-time measurements, feedback optimization still requires the Jacobian $x\in\mathbb{R}^n$3 (Huang et al., 20 Jul 2025).

The point of SLFO is therefore not merely to embed gradient descent in feedback, but to address the specific first-order-information bottleneck. Existing approximate feedback-optimization methods often use a fixed-point linearization, meaning that the plant is linearized once around a nominal operating point and that Jacobian approximation is reused throughout the optimization. The SLFO framework replaces that fixed approximation with a sequentially updated local one, so that the sensitivity information remains locally accurate as the iterate moves (Huang et al., 20 Jul 2025).

2. Sequential linearization mechanism

The local model at iterate $x\in\mathbb{R}^n$4 is

$x\in\mathbb{R}^n$5

Its steady-state sensitivity is computed from

$x\in\mathbb{R}^n$6

which yields the approximate Jacobian

$x\in\mathbb{R}^n$7

In the convergence analysis, the paper simplifies to $x\in\mathbb{R}^n$8, in which case

$x\in\mathbb{R}^n$9

and for the true steady-state map,

$u\in\mathcal U\subseteq \mathbb{R}^p$0

The last identity makes clear why updating the linearization point matters: the true steady-state Jacobian agrees with the local linearized sensitivity only when the linearization is taken at the relevant steady-state operating condition (Huang et al., 20 Jul 2025).

The sequential feedback-optimization iteration is

$u\in\mathcal U\subseteq \mathbb{R}^p$1

$u\in\mathcal U\subseteq \mathbb{R}^p$2

$u\in\mathcal U\subseteq \mathbb{R}^p$3

This is a projected gradient method in the primal variable $u\in\mathcal U\subseteq \mathbb{R}^p$4, but it is implemented in closed loop: actual plant output $u\in\mathcal U\subseteq \mathbb{R}^p$5 replaces the unknown steady-state output, while the Jacobian term is supplied by the local linearization at the current operating point (Huang et al., 20 Jul 2025).

Methodologically, SLFO is sequential in a precise sense. The linearization point evolves as

$u\in\mathcal U\subseteq \mathbb{R}^p$6

so the Jacobian approximation used at step $u\in\mathcal U\subseteq \mathbb{R}^p$7 is always

$u\in\mathcal U\subseteq \mathbb{R}^p$8

This is not sequential quadratic programming, repeated convexification of the original nonlinear optimization problem, or model predictive control over a finite prediction horizon. It is a projected feedback-gradient scheme whose defining ingredient is online local sensitivity refresh (Huang et al., 20 Jul 2025).

3. Assumptions, convergence, and multi-timescale updates

The convergence analysis assumes uniform contraction in state,

$u\in\mathcal U\subseteq \mathbb{R}^p$9

bounded and Lipschitz partial derivatives of $y\in\mathbb{R}^m$0, Lipschitz continuity of the steady-state map,

$y\in\mathbb{R}^m$1

bounded and Lipschitz partial gradients of $y\in\mathbb{R}^m$2, and strong monotonicity in $y\in\mathbb{R}^m$3,

$y\in\mathbb{R}^m$4

These conditions guarantee uniqueness of the optimal steady state and support a contraction-based analysis (Huang et al., 20 Jul 2025).

A key preliminary result is the Lipschitz estimate

$y\in\mathbb{R}^m$5

Another controls the plant-settling mismatch: $y\in\mathbb{R}^m$6 satisfies

$y\in\mathbb{R}^m$7

The asymptotic $y\in\mathbb{R}^m$8 tracking mismatch reflects a standard feedback-optimization effect: the optimizer updates while the plant is still converging to its steady state (Huang et al., 20 Jul 2025).

For the ideal feedback-optimization algorithm with exact $y\in\mathbb{R}^m$9, convergence is governed by the matrix

$w=\mathrm{col}\{w_1,w_2\}$0

with

$w=\mathrm{col}\{w_1,w_2\}$1

and

$w=\mathrm{col}\{w_1,w_2\}$2

If

$w=\mathrm{col}\{w_1,w_2\}$3

the ideal feedback-optimization algorithm converges to $w=\mathrm{col}\{w_1,w_2\}$4 (Huang et al., 20 Jul 2025).

For SLFO itself, the main practical-convergence bound is

$w=\mathrm{col}\{w_1,w_2\}$5

The neighborhood radius is therefore order $w=\mathrm{col}\{w_1,w_2\}$6. The paper explicitly interprets this as practical convergence rather than exact convergence: the iterates approach a neighborhood of the optimal steady state whose size is controlled by plant contraction, step size, and conditioning (Huang et al., 20 Jul 2025).

To reduce repeated linearization cost, the framework also introduces a multi-timescale variant. At outer iteration $w=\mathrm{col}\{w_1,w_2\}$7, the sensitivity

$w=\mathrm{col}\{w_1,w_2\}$8

is computed once, and then held fixed for $w=\mathrm{col}\{w_1,w_2\}$9 inner projected-gradient steps. The corresponding asymptotic bound becomes

$I-\nabla_x f(x,u)$0

When $I-\nabla_x f(x,u)$1, this reduces to the standard SLFO result; larger $I-\nabla_x f(x,u)$2 reduces linearization frequency but enlarges the ultimate bound (Huang et al., 20 Jul 2025).

4. Position within feedback optimization and feedback linearization

SLFO occupies a specific place inside a broader family of feedback-optimization and control-theoretic optimization methods. It is directly about steady-state feedback optimization for nonlinear systems, but it is distinct from several adjacent approaches.

A useful linear and sampled-computation special case is the hybrid framework for disturbed linear systems in which the plant evolves in continuous time, while optimization computations occur only at discrete update events (Chuy et al., 1 Apr 2025). That work models the closed loop as a hybrid system $I-\nabla_x f(x,u)$3, proves well-posedness, completeness, and absence of Zeno behavior, and shows exponential convergence to a ball of known radius around a desired fixed point. Its optimizer is a projected gradient-type iteration embedded in the loop, and the simulation study reports that the magnitude of steady-state error is $I-\nabla_x f(x,u)$4 less than the magnitude of the disturbances in the system (Chuy et al., 1 Apr 2025). The relationship to SLFO is not algorithmic but structural: it provides a mathematically explicit computation-in-the-loop template for analyzing finite-rate optimization interleaved with plant evolution.

A second neighboring line concerns constrained feedback optimization. The state-constrained method based on safe gradient flows and high-order control barrier functions formulates a nonlinear plant

$I-\nabla_x f(x,u)$5

and uses a quadratic program to modify a nominal gradient flow so that $I-\nabla_x f(x,u)$6 and $I-\nabla_x f(x,u)$7 hold for all time (Delimpaltadakis et al., 1 Apr 2025). This is explicitly not a sequential-linearization method: it does not repeatedly linearize the plant or solve sequential convexifications of the steady-state optimization problem. Its relevance to SLFO lies in transient safety rather than local-model updating.

A third neighboring line uses feedback linearization to solve constrained optimization by treating Lagrange multipliers as control inputs. In one formulation, exact feedback linearization of the constraint-output dynamics yields a continuous-time law whose forward-Euler discretization is equivalent to an SQP step when $I-\nabla_x f(x,u)$8 (Runyu et al., 16 Mar 2025). In a related continuous-time framework, the multiplier law

$I-\nabla_x f(x,u)$9

renders the constraint dynamics linear, $\phi$0, after which $\phi$1 enforces feasibility regulation (Cerone et al., 2024). These approaches are closer to control-theoretic sequential quadratic methods than to SLFO’s steady-state Jacobian approximation problem. They linearize constraint-output dynamics, whereas SLFO sequentially updates the local steady-state sensitivity used in a projected feedback-gradient step.

This contrast also clarifies a common misconception. SLFO is not classical feedback linearization, not fixed-linearization approximate feedback optimization, not data-driven sensitivity estimation by statistical learning from noisy excitation, and not real-time iteration MPC toward a known reference (Huang et al., 20 Jul 2025).

5. Wind farm control as a high-dimensional application

The flagship application of SLFO is wind farm control, where exact steady-state sensitivities are difficult to obtain in a high-dimensional nonlinear flow model (Huang et al., 20 Jul 2025). The application uses the WFSim model

$\phi$2

with power modeled as

$\phi$3

The steady-state optimization problem is

$\phi$4

This matches the generic SLFO setup: $\phi$5 is the large flow state, $\phi$6 is the control, $\phi$7 is the vector of turbine powers, and $\phi$8 is a power-tracking-plus-regularization objective (Huang et al., 20 Jul 2025).

Around a local operating point $\phi$9, the steady-state sensitivities are approximated by

$\phi(u,w_1)=f(\phi(u,w_1),u)+w_1,$0

$\phi(u,w_1)=f(\phi(u,w_1),u)+w_1,$1

These are application-specific instances of the generic local-gain formula $\phi(u,w_1)=f(\phi(u,w_1),u)+w_1,$2, and they are precisely the quantities whose sequential refresh is meant to preserve local accuracy as the operating point moves (Huang et al., 20 Jul 2025).

The reported numerical results illustrate both the performance and the computational motivation for the multi-timescale formulation. In the medium-fidelity OWEZ case, the greedy baseline achieves about $\phi(u,w_1)=f(\phi(u,w_1),u)+w_1,$3 MW, whereas SFO achieves about $\phi(u,w_1)=f(\phi(u,w_1),u)+w_1,$4 MW, corresponding to a $\phi(u,w_1)=f(\phi(u,w_1),u)+w_1,$5 improvement (Huang et al., 20 Jul 2025). For the slower-timescale variant, the reported computational data are: $\phi(u,w_1)=f(\phi(u,w_1),u)+w_1,$6: 1122 linearizations, 89760 forward simulations, 2400.01 s; $\phi(u,w_1)=f(\phi(u,w_1),u)+w_1,$7: 500 linearizations, 50000 forward simulations, 1178.08 s; $\phi(u,w_1)=f(\phi(u,w_1),u)+w_1,$8: 309 linearizations, 33990 forward simulations, 758.13 s (Huang et al., 20 Jul 2025). The monotone drop in linearization count and total runtime as $\phi(u,w_1)=f(\phi(u,w_1),u)+w_1,$9 increases is consistent with the theory’s explicit error-versus-computation trade-off.

High-fidelity validation also exposes the method’s sensitivity to model mismatch. In WFSim, SFO achieves a $y_{\text{ss}} = g(\phi(u,w_1),u)+w_2 =: h(u,w).$0 power gain with convergence. In SOWFA, stable convergence is lost due to model mismatch, but the moving average still shows a $y_{\text{ss}} = g(\phi(u,w_1),u)+w_2 =: h(u,w).$1 gain over greedy control (Huang et al., 20 Jul 2025). This is a direct empirical demonstration that sequential linearization improves over a fixed approximation but does not eliminate the consequences of inaccurate plant models.

6. Scope, limitations, and extensions

SLFO is best suited to nonlinear systems with a well-defined, stable equilibrium for each feasible input, sufficiently smooth dynamics, and access to derivative information rich enough to compute local Jacobians $y_{\text{ss}} = g(\phi(u,w_1),u)+w_2 =: h(u,w).$2 (Huang et al., 20 Jul 2025). Its strongest theoretical guarantees require uniform contraction in state and strong monotonicity of the cost gradient in the input variable. Those assumptions exclude many systems with weaker or only local stability properties.

Several limitations follow directly from the formulation. First, the method achieves practical convergence to a neighborhood, not exact convergence to the optimizer, and the bound depends on plant contraction and step size (Huang et al., 20 Jul 2025). Second, repeated re-linearization can be computationally expensive in large systems, which is why the multi-timescale variant is necessary in the first place (Huang et al., 20 Jul 2025). Third, sequential linearization remains model-based; if the local linearizations are built from a poor model, oscillations and degraded performance can persist, as the SOWFA study shows (Huang et al., 20 Jul 2025). Fourth, the base SLFO framework does not address transient safety constraints.

These limitations point toward several adjacent research directions already represented in the literature. The hybrid sampled-computation analysis of disturbed linear feedback optimization suggests how explicit timers, event schedules, and plant-optimizer interleaving can be analyzed rigorously when computation and plant motion live on different time domains (Chuy et al., 1 Apr 2025). The HOCBF-based safe feedback-optimization formulation suggests one route to enforcing state and input constraints at all times, although that route does not use sequential linearization (Delimpaltadakis et al., 1 Apr 2025). A plausible implication is that future SLFO variants could combine online local sensitivity updates with a QP-based safety layer or with hybrid sampled-data analysis, but such combinations are not developed in the cited works.

In the current literature, SLFO is therefore best understood as a model-based, locally adaptive Jacobian-approximation framework for steady-state feedback optimization. Its distinctive contribution is not merely the use of local linear models, but the decision to move the linearization point with the optimization trajectory and to quantify the resulting asymptotic neighborhood explicitly (Huang et al., 20 Jul 2025).