A Hybrid Systems Model of Feedback Optimization for Linear Systems: Convergence and Robustness

1. Problem Statement

Feedback optimization (FO) is a control strategy that computes system inputs in real-time using output measurements to optimize steady-state behavior, rather than relying on pre-computed reference signals. While effective, existing literature typically models FO in one of two ways:

1. Purely Continuous-Time: Both system dynamics and optimization algorithms are continuous.
2. Purely Discrete-Time: Both system dynamics and optimization algorithms are discrete.

The Gap: Practical implementations often involve continuous-time physical systems driven by discrete-time digital computers (sampled-data systems). Existing models do not accurately capture the dynamics of this hybrid nature, particularly regarding the timing of input updates, output sampling, and the computational time required for optimization iterations. Furthermore, there is a lack of rigorous analytical proofs regarding the robustness of such hybrid FO systems to simultaneous perturbations in model parameters, measurement noise, and timing irregularities.

Objective: The authors aim to formulate a rigorous hybrid systems model for feedback optimization that unifies continuous-time plant dynamics with discrete-time optimization computations. They seek to prove the well-posedness of this model, establish convergence to a desired state, and analytically demonstrate robustness against various disturbances.

2. Methodology

A. System Formulation

The authors model a Linear Time-Invariant (LTI) plant:
$$\dot{x} = Ax + Bu, \quad y = \Psi x + d$$
where $x$ is the state, $u$ is the input, $y$ is the output, and $d$ is an unknown constant disturbance. The goal is to minimize a strongly convex objective function $\Phi(u, y)$ (specifically quadratic in this work) subject to constraints.

B. Hybrid System Construction

The core contribution is the formulation of the closed-loop system as a Hybrid System $\mathcal{H}_{FO} = (C, F, D, G)$ in the framework of Goebel, Sanfelice, and Teel [18]. The state vector $\zeta$ includes:

• $x, u$: Plant state and input.
• $y_s$: Sampled output (held constant between samples).
• $z$: Current iterate of the gradient descent algorithm.
• $\tau_c, \tau_g$: Timers tracking time until the next input update and time remaining for a computation step, respectively.

Dynamics:

• Flow Set ($C$): The system flows (continuous evolution) while timers $\tau_c > 0$ and $\tau_g > 0$. During flow, the plant state evolves according to $\dot{x} = Ax + Bu$, while inputs and optimization iterates remain constant. Timers decrease at a unit rate.
• Jump Set ($D$): Jumps occur when $\tau_c = 0$ (input update/sampling) or $\tau_g = 0$ (computation step).
• Jump Map ($G$):
  • Case 1 ($\tau_g=0, \tau_c>0$): Perform one gradient descent step to update $z$. Reset $\tau_g$.
  • Case 2 ($\tau_c=0, \tau_g>0$): Apply the latest $z$ as the new input $u$, sample the new output $y_s$, and reset $\tau_c$ to a random value in $[\tau_{c,min}, \tau_{c,max}]$.
  • Case 3 ($\tau_c=0, \tau_g=0$): Simultaneous execution of both updates.

C. Theoretical Analysis Tools

The authors utilize tools from hybrid systems theory to analyze:

1. Well-posedness: Ensuring solutions exist and are unique (in a set-valued sense).
2. Completeness: Ensuring solutions do not terminate in finite time.
3. Zeno Behavior: Proving the system does not undergo infinite jumps in finite time.
4. Robustness: Using the concept of $(\tau, \epsilon)$-closeness to show that perturbed trajectories remain close to nominal trajectories over bounded time horizons.

3. Key Contributions

1. Hybrid Modeling of Feedback Optimization:
   The paper presents the first hybrid model that explicitly captures the interplay between continuous plant dynamics, discrete-time optimization iterations, and aperiodic sampling/input updates. This model allows for non-periodic sampling and variable computation times.

2. Well-Posedness and Non-Zeno Properties:
   The authors prove that the proposed hybrid model satisfies the "Hybrid Basic Conditions," ensuring it is well-posed. They further prove that all maximal solutions are complete (exist for all time) and non-Zeno (no infinite jumps in finite time), ruling out pathological behaviors.

3. Convergence Analysis:

   • Theorem 1 & 2: The paper establishes that the system state converges exponentially to a ball of known radius around the desired optimal steady-state.
   • The size of this residual ball depends on system parameters, the step size of the gradient descent, and the minimum time between input updates.
   • The analysis holds for both specific initial conditions and globally (arbitrary initial conditions).

4. Analytical Robustness Guarantees:

   • Theorem 3: This is a novel contribution. The authors prove that the hybrid FO system is robust to perturbations over bounded time horizons.
   • The robustness covers three simultaneous types of disturbances:
     1. Errors in measured output values.
     2. Errors in the LTI system matrices ($A, B, \Psi$).
     3. Errors in the timing of input applications (sampling jitter).
   • This is the first analytical proof showing simultaneous robustness to all these specific hybrid disturbances.

4. Results

• Simulation Validation: The authors simulated a 4-state LTI system with a quadratic objective.
  • Nominal Case: The state $x$ converged to the optimal steady-state $\tilde{x}$ with a negligible error ($\approx 10^{-15}$), confirming exponential convergence.
  • Perturbed Case: The system was subjected to perturbations in model matrices ($\hat{A}, \hat{B}, \hat{H}$) and timer rates ($\kappa$).
  • Findings: The simulations confirmed that while perturbations increased the steady-state error, the system remained stable. The error magnitude was found to be primarily dependent on the magnitude of model matrix perturbations ($\mu$) rather than timing errors ($\kappa$), though timing errors affected convergence rates.
• Quantitative Bounds: The derived bounds (Theorem 1) successfully predicted the radius of the convergence ball, validating the theoretical analysis.

5. Significance

• Bridging Theory and Practice: By modeling the system as a hybrid entity, the paper provides a theoretical framework that accurately reflects real-world digital control implementations where continuous plants are controlled by discrete algorithms.
• Rigorous Robustness: Unlike prior works that often rely on simulations or assume idealized periodic sampling, this paper provides analytical proofs of robustness against a comprehensive set of realistic disturbances (modeling errors, measurement noise, and timing jitter).
• Foundation for Future Work: The framework established here (using the hybrid systems formalism of [18]) opens the door for analyzing more complex scenarios, such as non-convex objectives and nonlinear dynamics, which are currently beyond the scope of standard feedback optimization literature.
• Safety and Reliability: The proof of non-Zeno behavior and completeness is crucial for safety-critical applications, ensuring the controller will not "freeze" or behave unpredictably due to rapid switching.

In summary, this paper advances the field of feedback optimization by providing a mathematically rigorous, hybrid-systems-based model that guarantees stability, convergence, and robustness in practical, sampled-data environments.