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On the design of stabilizing FIR controllers

Janis Adamek, Nils Schlüter, Moritz Schulze Darup

TL;DR

This paper studies stabilizing finite impulse response (FIR) controllers for linear discrete-time plants, motivated by encrypted control where FIR controllers are advantageous. It shows FIR design can be reframed as static or dynamic output feedback, tying existence to strong stabilizability and the parity interlacing property, and demonstrates that classical convex LMIs become overly restrictive for FIR structures. Through theoretical results and numerical case studies, the work reveals that convex FIR design methods often fail (and increasing FIR order does not guarantee feasibility), while non-convex optimization can yield satisfactory FIR controllers, albeit with higher computational costs. The findings highlight the practical relevance of non-convex design for FIR controllers in privacy-preserving control and point to future work on expanding classical designs under FIR constraints and applying FIRs as low-level primitives in encrypted control systems.

Abstract

Recently, it has been observed that finite impulse response controllers are an excellent basis for encrypted control, where privacy-preserving controller evaluations via special cryptosystems are the main focus. Beneficial properties of FIR filters are also well-known from digital signal processing, which makes them preferable over infinite impulse response filters in many applications. Their appeal extends to feedback control, offering design flexibility grounded solely on output measurements. However, designing FIR controllers is challenging, which motivates this work. To address the design challenge, we initially show that FIR controller designs for linear systems can equivalently be stated as static or dynamic output feedback problems. After focusing on the existence of stabilizing FIR controllers for a given plant, we tailor two common design approaches for output feedback to the case of FIR controllers. Unfortunately, it will turn out that the FIR characteristics add further restrictions to the LMI-based approaches. Hence, we finally turn to designs building on non-convex optimization, which provide satisfactory results for a selection of benchmark systems.

On the design of stabilizing FIR controllers

TL;DR

This paper studies stabilizing finite impulse response (FIR) controllers for linear discrete-time plants, motivated by encrypted control where FIR controllers are advantageous. It shows FIR design can be reframed as static or dynamic output feedback, tying existence to strong stabilizability and the parity interlacing property, and demonstrates that classical convex LMIs become overly restrictive for FIR structures. Through theoretical results and numerical case studies, the work reveals that convex FIR design methods often fail (and increasing FIR order does not guarantee feasibility), while non-convex optimization can yield satisfactory FIR controllers, albeit with higher computational costs. The findings highlight the practical relevance of non-convex design for FIR controllers in privacy-preserving control and point to future work on expanding classical designs under FIR constraints and applying FIRs as low-level primitives in encrypted control systems.

Abstract

Recently, it has been observed that finite impulse response controllers are an excellent basis for encrypted control, where privacy-preserving controller evaluations via special cryptosystems are the main focus. Beneficial properties of FIR filters are also well-known from digital signal processing, which makes them preferable over infinite impulse response filters in many applications. Their appeal extends to feedback control, offering design flexibility grounded solely on output measurements. However, designing FIR controllers is challenging, which motivates this work. To address the design challenge, we initially show that FIR controller designs for linear systems can equivalently be stated as static or dynamic output feedback problems. After focusing on the existence of stabilizing FIR controllers for a given plant, we tailor two common design approaches for output feedback to the case of FIR controllers. Unfortunately, it will turn out that the FIR characteristics add further restrictions to the LMI-based approaches. Hence, we finally turn to designs building on non-convex optimization, which provide satisfactory results for a selection of benchmark systems.
Paper Structure (8 sections, 5 theorems, 39 equations, 1 figure)

This paper contains 8 sections, 5 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem specification and reformulations
  3. Existence of stabilizing FIR controllers
  4. No free lunch from classical approaches
  5. Convexification for static output feedback
  6. Convexification for dynamic output feedback
  7. Numerical Analysis
  8. Conclusion and Outlook

Key Result

Lemma 1

The matrices eq:closedLoopAugmented and eq:closedLoopDynamic specified by eq:augmentedSystemMatrices, eq:uOutputFeedbackAugmentedY, and eq:controllerParameters are equal.

Figures (1)

  • Figure 1: Median closed-loop spectral radius for increasing FIR controller order (HiSyn toolbox in solid, GA in dashed lines). Shaded areas show deviations for different starting values.

Theorems & Definitions (9)

  • Lemma 1
  • Proof 1
  • Theorem 1
  • Lemma 2
  • Proof 2
  • Lemma 3
  • Proof 3
  • Lemma 4
  • Proof 4