Quasi-Static Feedback Strategy

• Quasi-static feedback strategy is a control method that replaces high-order, nonphysical state feedback with locally measurable, low-order states for effective error decoupling.
• It enables exact linearization and robust tracking in flat, underactuated, and hybrid systems by leveraging algebraic reparameterization and constructive feedback laws.
• The approach improves analytical tractability and performance optimization under uncertainty, with applications spanning robotics, wind-farm control, and communication systems.

A quasi-static feedback strategy is a class of control or estimation law characterized by the (local, often algebraic) replacement of generalized or high-order state feedback—typically in the context of flat systems or systems with special geometric structure—by feedback involving only classical, measurable, or low-differentiation states. This approach is motivated by the desire to achieve desirable closed-loop properties (such as exact error decoupling, linear error dynamics, or robust performance under feedback) while avoiding the need to estimate or differentiate generalized states involving fictitious or high-order variables that are nonphysical or poorly observed. The quasi-static feedback paradigm permeates a range of fields, including nonlinear control of flat and underactuated systems, non-prehensile manipulation under uncertainty, wind-farm closed-loop power/load optimization, as well as tracking in stochastic and hybrid systems, each with domain-specific instantiations.

1. Quasi-Static Feedback in Flat System Control

In control theory, flatness refers to the property that the system’s state and input can be parameterized in terms of a so-called flat output and its time-derivatives. For $(x,u)$-flat systems—where the flat output is a function of the state and input but not derivatives of the input—a central concern is the synthesis of decoupled, linear, and asymptotically stable error dynamics for output tracking without appealing to the generalized Brunovský state, whose components typically require high-order differentiation and are not physically measurable (Gstöttner et al., 2021).

Given a system $\dot{x} = f(x, u)$ and a flat output $y = \varphi(x, u)$, the classical flatness-based tracking law proceeds by:

1. Expressing the system as a chain of decoupled integrators by formal differentiation of the flat output.
2. Designing a static feedback linearizing law in the generalized coordinates.
3. Assigning the closed-loop error dynamics (typically via pole-placement).

However, practical realization motivates feedback laws that only require measurement of $x$ and known references. The quasi-static feedback strategy achieves this by reparameterizing the tracking law as

$u = \alpha\left(x, y^d_{[0, R]}\right),$

where $y^d$ is the reference trajectory, and $R$ is the minimal flatness index, based solely on derivatives of the reference and the classical state $x$.

Subject to a rank condition on a suitable multi-index $\kappa$ (with $|\kappa|=n$ and $\kappa \leq R$), the necessary linearizing feedback is found via a lower-triangular constructive algorithm, yielding a mapping $u = \bar F_u(x, v_{[0, R-\kappa]})$ where the virtual input $v$ replaces otherwise unmeasurable generalized derivatives. Linear tracking error dynamics in the outputs $y$ are then achieved through an outer static feedback assignment

$$v^{j}_{[\kappa^j]} + \sum_{\beta=0}^{\kappa^j-1} a_{j,\beta} e^j_{[\beta]} = 0.$$

This construction fully circumvents the need for estimation of generalized states and guarantees decoupled, asymptotically stable error convergence, as shown in detailed case studies with high-dimensional academic systems and gantry cranes (Gstöttner et al., 2021).

2. Exact Linearization in Minimally Underactuated Lagrangian Systems

For minimally underactuated configuration-flat Lagrangian systems—systems with $p$ degrees of freedom and $m=p-1$ inputs—the quasi-static feedback paradigm enables exact linearization of the closed-loop system into decoupled integrator chains via feedback in the classical (configuration and velocity) state, entirely avoiding generalized coordinate transforms (Hartl et al., 2023).

The strategy exploits a local diffeomorphism between the configuration, velocities, and a flat output $y=\phi(q)$ and a complementary coordinate. Through judicious choice of a vector $\kappa$ of integrator-chain lengths, and leveraging implicit function theorem arguments, one constructs a feedback

$$u = \Phi\bigl(q, v, w, w_{[1]}, ..., w_{[R-\kappa]}\bigr)$$

where $w^j := y^j_{[\kappa^j]}$ serve as virtual inputs. Under this law, the system implements $m$ chains of integrators of prescribed length, and by minimality of $\kappa$—one long, rest-to-rest compatible chain and $m-1$ short chains—ensures well-posedness of the feedback and regularity near equilibria.

Illustrative examples, such as the planar VTOL, show explicit realization of these laws, yielding regular feedback at hover and rest configurations, confirming the method's ability to avoid singularities otherwise introduced by non-minimal chain lengths (Hartl et al., 2023).

3. Application to Non-Prehensile Manipulation under Uncertainty

In non-prehensile robotic manipulation, where contact dynamics are subject to high uncertainty and non-smooth hybrid effects, quasi-static feedback strategies exploit the fact that under high-stiffness and quasi-static assumptions, the coupled robot–object system can be reduced to deterministic and analytically tractable maps. All contact-induced uncertainty is lumped into stochastic perturbations activated only upon contact (Jankowski et al., 3 Apr 2024).

Crucially, the method propagates the belief (mean and covariance) of the object’s configuration analytically—bypassing computationally intractable belief evolution—by: $$\text{Var}[q^{\mathrm{o}}_{+}] = \text{Var}[f(q^{\mathrm{o}}, u)] + \mathbb{E}[\eta] V_w,$$ where $f(q^{\mathrm{o}}, u)$ denotes the deterministic contact map, $\eta$ is the binary contact random variable, and $V_w$ the contact noise covariance.

Robust open-loop plan generation employs a variance-gain constraint on the growth of covariance: $\gamma_k = \frac{V_{k+1}}{V_k + V_w} \leq 1$ and enforces this within a sampling-based trajectory optimizer via a cost barrier. The overall approach is then embedded in a model-predictive (receding-horizon) feedback loop, updating the belief with real-world particle rollouts after short plan executions. The combination of analytical variance propagation, robust plan constraints, and receding-horizon correction typifies the quasi-static feedback strategy for robust non-prehensile manipulation (Jankowski et al., 3 Apr 2024).

4. Cascaded Quasi-Static Feedback in Flat Mechanical Systems

Differentially flat slider-pusher systems under quasi-static assumptions (negligible inertia, high support friction, negligible pusher–slider friction) admit cascaded quasi-static feedback controllers that sequentially regulate outer position, mid-level heading, and inner contact offset error loops. Each sub-controller is realized via a local inversion of the analytic quasi-static kinematic model, closing error dynamics with time-scale separation (Witte et al., 6 Nov 2025).

For a rectangular slider with circular pusher, the state vector combines position, heading, and tangent offset. A flat output is identified with the center-of-mass position, from which all states and necessary inputs can be algebraically computed. The cascaded loops—outer loop: position velocity, mid-loop: heading, inner loop: contact offset—implement error-regulation via linear controllers and analytic model inversion at each stage. This structure ensures bounded tracking errors under modeling uncertainty and input noise, as validated by both hardware and high-fidelity simulation (Witte et al., 6 Nov 2025).

5. Quasi-Static Feedback for Performance Optimization under Constraints

Quasi-static feedback strategies are effectively employed for performance optimization in systems exhibiting slow-varying, quasi-steady dynamics, for instance in wind-farm control (Sood et al., 2023). Here, turbine setpoints (yaw angles, pitch) are adjusted in a receding-horizon (interval-by-interval) closed-loop to optimize combined power and fatigue objectives, with the underlying flow and wake fields assumed quasi-static over each optimization interval.

The methodology comprises three key steps:

1. Analytical, parameterized wake modeling and offline fatigue lookup table construction for rapid surrogate evaluation.
2. Periodic closed-loop calibration: at the end of each interval, new measurements are assimilated to re-estimate the ambient wind and recalibrate wake parameters using SCADA data.
3. Single-shot optimization of setpoints subject to constraints (e.g., yaw limits) and reapplication on the plant, closing the feedback loop.

The strategy demonstrates significant performance gains, robust adaptation to dynamic operational changes (e.g., turbine shut-down), and effectiveness under practical fatigue constraints (Sood et al., 2023).

6. Quasi-Static State-Feedback in Stochastic and Hybrid Tracking

In stochastic tracking problems with both continuous (diffusive) and impulsive (jump) controls, quasi-static feedback refers to stationary state-dependent laws maintaining the deviation within a time-varying domain via regular drift and boundary resets. For Itô process targets, the optimal quasi-static feedback preserves the deviation inside a bounded region using a drift $U_t$ and jump map $\epsilon_t$ chosen so the empirical law of the deviation converges to the solution of a time-average linear program (LP) for cost minimization (Cai et al., 2016). The approach is shown to be asymptotically optimal in the small-cost regime for a broad class of examples, provided certain potential and separability conditions on the domain and feedback mappings are satisfied.

7. Quasi-Static Feedback in Communication Systems

In the context of communication over quasi-static fading channels, the notion of quasi-static feedback underpins several strategies involving quantized channel state feedback and/or feedback-assisted coding. For instance, in blocklength-limited goodput maximization with quantized receiver feedback, the transmitter updates its coding rate according to the quantized channel state index, with feedback decisions held fixed within each block—hence the action is "quasi-static" in both physical and control space (Celebi et al., 2022). Similarly, the coding design for quasi-static fading channels with imperfect channel state information at the transmitter (I-CSIT) and quantized feedback introduces feedback quantization constraints and uses auxiliary signals (modulo-lattice maps) to eliminate residual error propagation, again leveraging the quasi-static nature of the channel within each block and of the feedback response (Yang et al., 2 Jul 2025).

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Through these diverse instantiations, the quasi-static feedback strategy is consistently characterized by (i) the exploitation of local, low-order, or algebraic relations in state-feedback design, (ii) the circumvention of high-order or generalized state estimation, (iii) analytic tractability and formal guarantees (decoupling, optimality, robustness) under suitable regularity and rank conditions, and (iv) demonstrated practical efficacy in domains ranging from nonlinear geometric control to robust manipulation and real-time optimization under constraints.