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

Feedback optimization algorithms compute inputs to a system using real-time output measurements, which helps mitigate the effects of disturbances. However, existing work often models both system dynamics and computations in either discrete or continuous time, which may not accurately model some applications. In this work, we model linear system dynamics in continuous time, and we model the computations of inputs in discrete time. Therefore, we present a novel hybrid systems model of feedback optimization. We first establish the well-posedness of this hybrid model and establish completeness of solutions while ruling out Zeno behavior. Then we show the state of the system converges exponentially fast to a ball of known radius about a desired goal state. Next we analytically show that this system is robust to perturbations in (i) the values of measured outputs, (ii) the matrices that model the linear time-invariant system, and (iii) the times at which inputs are applied to the system. Simulation results confirm that this approach successfully mitigates the effects of disturbances.

💡 Research Summary

The paper addresses a gap in the feedback‑optimization literature where system dynamics and the optimization algorithm are typically modeled either both in continuous time or both in discrete time. In practice, physical plants evolve continuously while the controller runs on a digital processor that updates inputs at discrete instants. To capture this mixed nature, the authors develop a hybrid‑systems model that couples a continuous‑time linear time‑invariant (LTI) plant with a discrete‑time gradient‑descent optimizer.

The plant is described by (\dot x = A x + B u,; y = \Psi x + d) with a Hurwitz matrix (A) and an unknown constant disturbance (d). The optimizer seeks to minimize a strongly convex objective (\Phi(u,y)) subject to the steady‑state relation (y = H u + d) where (H = -\Psi A^{-1} B). Because (d) and the plant matrices are not known exactly, the algorithm uses sampled measurements (y_s) taken at irregular times. The discrete update rule is (u_{k+1}= \Pi_U\bigl(u_k - \gamma \nabla_u \Phi(u_k, y_s)\bigr)).

The hybrid state consists of the plant state (x), the current input (u), the most recent sampled output (y_s), the optimizer iterate (z), and two timers (\tau_c) (time until the next input change) and (\tau_g) (time required to complete a gradient step). The flow set (C) contains states with (\tau_c>0) and (\tau_g>0); the flow map (F) captures the continuous plant dynamics and the linear decay of the timers. The jump set (D) is triggered when either timer reaches zero, and the jump map (G) implements three cases: (i) a gradient step finishes ((\tau_g=0)) but the input is not yet updated, (ii) the input update time expires ((\tau_c=0)) and the new optimizer iterate is applied, and (iii) both events occur simultaneously.

The authors first prove that the hybrid data ((C,F,D,G)) satisfy the Hybrid Basic Conditions, guaranteeing well‑posedness (existence of solutions). They then show that Zeno behavior cannot occur because each jump is separated by a positive dwell time, and all maximal solutions are complete.

The convergence analysis demonstrates exponential convergence of the hybrid trajectories to a ball around the optimal equilibrium ((x^\star,u^\star)). The radius of this ball is explicitly expressed in terms of the step size (\gamma), the strong convexity constants of (\Phi), the disturbance magnitude, and the eigenvalues of (A). Thus, even with constant disturbances, the system settles arbitrarily close to the optimum by choosing appropriate algorithm parameters.

Robustness is the centerpiece of the work. The authors consider three simultaneous perturbation classes: (i) measurement noise affecting the sampled output (y_s), (ii) parametric errors in the plant matrices (A,B,\Psi), and (iii) timing errors in the sampling and input‑update instants. By leveraging the hybrid‑systems framework, they prove that over any bounded hybrid time horizon (