Internal Model Principle in Feedback Optimization

• Internal Model Principle in feedback optimization is a design strategy that embeds models of exogenous disturbances within controllers to achieve exact tracking of time-varying optima.
• It integrates static and dynamic control designs using output regulation and observer-based techniques to replicate disturbance dynamics in real time.
• This approach removes the need for strict time-scale separation, ensuring robust, high-precision control as demonstrated in simulation studies on dynamic systems.

Feedback optimization in dynamical systems is fundamentally influenced by the internal model principle—a concept originating in robust regulation theory stating that, for perfect tracking or disturbance rejection, the controller must internally reproduce the dynamics of external disturbances or reference signals. In modern feedback optimization, especially for time-varying optimization problems, this principle dictates that the control algorithm must include a mathematical model of the time variability affecting the optimal operation point, enabling exact tracking of moving optima even in the presence of rapid and structured temporal variations(Bianchin et al., 5 Aug 2025).

1. Formal Statement of the Internal Model Principle in Feedback Optimization

The internal model principle (IMP) applied to feedback optimization asserts that a controller can track the time-varying optimizer of an optimization problem—subject to exogenous disturbances—if and only if it internally contains a model of the exogenous disturbance dynamics. For a plant influenced by a time-varying disturbance $w(t)$, whose evolution is generated by an exosystem

$\dot{w}(t) = s(w(t)),$

and an associated optimization objective $\phi(u, w(t))$ with critical point $u^*(t)$, the IMP states that the controller must replicate (internally) the dynamics $s(\cdot)$ of the exosystem for exact zero-error tracking. Mathematically, the controller state $z(t)$ must satisfy a coordinate transformation $z = \sigma(w)$ so that its own dynamics evolve in parallel with those of the disturbance generator, as formally described in the necessary and sufficient conditions on the controller structure such as equations (dyn_feedback_a), (dyn_feedback_b), and (dyn_feedback_c) in (Bianchin et al., 5 Aug 2025).

2. Output Regulation Perspective and Design Architecture

By reframing feedback optimization as an output regulation problem, the task of tracking a time-varying optimizer becomes one of regulating the gradient signal

$g(t) = \nabla_u \phi(u(t), w(t))$

to zero despite $\dot{w}(t) \neq 0$. The controller is thus structured as follows:

• Static component: Generates control actions as $u(t) = H_c(x(t), w(t)) = \gamma(w) + K(x - \pi(w))$, where $\pi(w)$ and $\gamma(w)$ solve the regulator equations:

$$\frac{\partial \pi}{\partial w} s(w) = f(\pi(w), \gamma(w), w), \qquad \nabla_u \phi(\gamma(w), w) = 0.$$

• Dynamic/observer component: If the plant state $x(t)$ or disturbance $w(t)$ is not directly measured, a dynamic observer (typically a Luenberger observer or a more general observer designed using system-theoretic tools) is synthesized to estimate both, using measured outputs $y(t)$ and known plant and exosystem models.

This separation principle—partitioning the controller into a regulation law (for driving the system toward the optimizer) and an internal model (for replicating disturbance dynamics)—enables the design of output-feedback controllers capable of exact optimization error regulation even as the optimizer varies with time.

3. The Necessity of Internal Models for Fast and Robust Tracking

The core limitation of classical feedback optimization schemes—either numerical optimization algorithms used in the loop or those relying on time-scale separation—is their inability to track rapidly-varying optima unless the controller operates on a much slower time-scale than the plant. The internal model principle, as formalized in (Bianchin et al., 5 Aug 2025), overcomes this by fundamentally removing this limitation: the presence of an internally-replicated exosystem dynamics allows the controller to synchronize with the time variations of the optimizer, thereby:

• removing the requirement for slow controller updates,
• enabling arbitrarily fast tracking (subject to physical/observer limitations), and
• guaranteeing exact regulation for all disturbances admitting a smooth exosystem representation.

Theoretical foundations, such as Theorem 4.4 and related results, provide necessary and sufficient conditions for this internal model structure to exist and be implemented in the controller, based on system smoothness, observability, and stabilizability.

4. Controller Synthesis: Static and Dynamic Designs

4.1 Static-Feedback Optimization

When the full system state and disturbance are measurable, the static-feedback controller writes as: $$u(t) = H_c(x(t), w(t)) = \gamma(w) + K(x - \pi(w)),$$ where

$$\frac{\partial \pi}{\partial w} s(w) = f(\pi(w), \gamma(w), w), \qquad \nabla_u \phi(\gamma(w), w) = 0.$$

Here, the mappings $\pi(w)$ and $\gamma(w)$ respectively parametrize the optimal invariant manifold and associated optimizer as a function of the disturbance $w$.

4.2 Dynamic-Feedback Optimization

For the more general (and practical) scenario when $x(t)$ and $w(t)$ are not directly available, an observer-based architecture is constructed in accordance with Algorithm 1 of (Bianchin et al., 5 Aug 2025). Specifically, the controller state $z(t)$ is composed of copies of the internal model and plant state estimator, and the controller equations take the form: $\dot{z} = F_c(z, y),$

$u = G_c(z),$

where $F_c$ includes the internal model of the exosystem, a state estimator for the plant, and appropriate correction terms involving output measurements.

5. Advantages Over Classical Feedback Optimization

Compared to classical numerical optimization-based feedback schemes—which often require strong separation between plant and controller time scales, and typically only guarantee approximate tracking of time-varying optimizers—the internal model principle-based approach provides the following improvements:

• Exact asymptotic tracking: Gradient error $g(t)$ is driven to zero even for non-constant, structured time variations in the disturbance.
• No strict time-scale separation: Stability and performance are achieved even when the controller operates at a rate comparable to (or faster than) the plant.
• Explicit performance guarantees: The necessary and sufficient conditions for exact tracking are formalized and constructive, enabling certification of performance for a broad class of time-varying problems.

These improvements arise from the precise embedding of the exogenous dynamics in the controller’s state evolution, as demanded by the internal model principle for feedback optimization.

6. Practical Examples and Case Studies

Simulation studies in (Bianchin et al., 5 Aug 2025) include balancing control of a Segway-like robot under exogenous time-varying disturbances, with cost functions ranging from quadratic forms to logistic losses. The dynamic-feedback controller, designed according to the proposed internal model methodology, regulates the gradient error to within numerical precision (often $< 10^{-6}$), ensuring robust upright stabilization and tracking across a range of disturbance strengths and time profiles. These results contrast with standard feedback optimization methods, which exhibit higher residual error or require slow controller updates.

7. Conclusion and Impact

The internal model principle, as formalized for feedback optimization in time-varying environments, provides a powerful, output-regulation–inspired framework for designing controllers capable of tracking the optimizer of time-varying objectives with high precision and speed. By requiring the embedding of explicit models of the disturbance dynamics within the controller, it provides both a necessary and sufficient condition for exact optimization in dynamic settings. This perspective not only generalizes the scope of feedback optimization beyond time-invariant or slowly-varying environments but also establishes a rigorous foundation for combining disturbance modeling, output feedback stabilization, and dynamic observer design in the synthesis of high-performance feedback optimization controllers for modern dynamical systems(Bianchin et al., 5 Aug 2025).