PaperlayerView exact revision on arXiv

Source-linked AI summary

A Frequency Domain Approach to Bounding Riccati Equation Perturbations

Rachel Newton, Laura Balzano, Peter Seiler

arXiv:2609.38515v1eess.SYcs.SYmath.OC

TL;DR

Uncertain LQR data make it useful to bound how perturbations affect DARE solutions. The paper derives an explicit frequency-domain bound using a closed-loop stability measure and weaker detectability assumptions; the resulting bound is linear in ϵ for sufficiently small perturbations.

  • Problem

    The paper addresses how perturbations to LQR data affect the DARE solutions, for which explicit bounds are useful.

    Source: §I · p. 1

  • Method

    The approach uses a frequency-domain closed-loop stability measure and relates LQR cost to the H2 norm to derive a computable DARE-solution bound.

    Source: §I · p. 1, §III · p. 2, §IV · p. 4

  • Results

    The DARE-solution difference has an explicit bound for sufficiently small ϵ, and the bound is linear in ϵ.

    Source: §IV · pp. 4–5

  • Takeaways & Limitations

    The bound uses weaker detectability assumptions, permits positive semidefinite DARE solutions, and applies in more general situations.

    Source: §I · p. 1, §IV · p. 6

Abstract

from arXiv · show

The discrete algebraic Riccati equation (DARE) is used to solve for the optimal feedback gain in linear-quadratic regulator (LQR) control of discrete-time systems. In this letter, we consider the effect of perturbations to the LQR data on the DARE solutions. We present a new frequency domain approach to derive an explicit bound on the difference in DARE solutions under such perturbations. Our bounds are similar to other results that have recently appeared in the literature with two distinctions. First, we use a frequency domain constant that measures the degree of closed-loop stability, in contrast to other results that use a time domain measure. Second, we start from weaker detectability assumptions on the LQR state cost matrix.

I. INTRODUCTION

Because LQR system data may be uncertain, the paper derives computable bounds on how perturbations affect DARE solutions. Its frequency-domain approach uses a closed-loop stability measure and weaker detectability assumptions than prior bounds.

I. INTRODUCTION

  • Prior DARE perturbation bounds use operator-theoretic or self-bounding ODE approaches, with some results extending to certain time-varying systems.One cited operator-theoretic result gives a tighter bound under stronger assumptions on the LQR data.Source: §I · p. 1
  • The proposed approach derives a computable bound using the frequency-domain closed-loop stability measure ˆcA1 rather than the time-domain measure τ used in existing bounds.The paper defines ˆcA1 in (28).Source: §I · p. 1
  • The bound assumes Q is positive semidefinite and (C, A) is detectable, allowing DARE solutions that are positive semidefinite.The paper contrasts these assumptions with prior assumptions that Q is positive definite or satisfies an observability assumption.Source: §I · p. 1
  • The proof framework bounds the difference between DARE solutions by first bounding differences in closed-loop H2 norms and using their equivalence to LQR cost.The paper outlines this sequence as its route to proving the DARE solutions are close.Source: §I · p. 1

II. PROBLEM FORMULATION

The paper formulates stochastic LQR for discrete-time systems and asks how perturbations to dynamics and state-cost factors affect the DARE solution. It seeks an explicit bound for sufficiently small perturbations under stabilizability and detectability assumptions.

II. PROBLEM FORMULATION

  • The LQR problem uses static state feedback uk = −Kxk to minimize an infinite-horizon quadratic cost, with Q = C⊤C and R specifying the state-regulation and control-effort trade-off.The state and input follow discrete-time dynamics, with i.i.d. zero-mean Gaussian process noise of covariance Σw.Source: §II · p. 1
  • The setup assumes (A, B) is stabilizable and (C, A) is detectable, with Q positive semidefinite and R positive definite.Under the stated assumptions, the DARE has a unique positive semidefinite solution.Source: §II · pp. 1–2
  • The perturbation question compares two LQR problems sharing Σw and R, with differences in A, B, and C bounded by ϵ.The state-cost perturbation is expressed through the square-root factors C1 and C2; perturbing R is excluded for simplicity.Source: §II · p. 2
  • The goal is an explicit constant c1, expressed using only the first LQR problem's data, that bounds the DARE-solution difference for sufficiently small ϵ.The passage states that DARE solutions vary continuously with problem data and that the bound quantifies this closeness.Source: §II · p. 2
  • The process-noise covariance is shared because it affects optimal LQR cost but does not enter the DARE or the perturbation analysis.The stated assumptions on both problems can be weakened to require them only for Problem 1 when ϵ is sufficiently small.Source: §II · p. 2

III. PERTURBATION BOUNDS ON THE H2-NORM

The paper relates stochastic LQR cost to the H2 norm of a closed-loop system, then bounds how that norm changes when system matrices are perturbed. This provides an intermediate step toward bounding differences in DARE solutions.

III. PERTURBATION BOUNDS ON THE H2-NORM

  • The Hardy space H2 consists of stable systems that are square-integrable on the unit circle, and its norm is used to represent the steady-state output measure.The section introduces the transfer function's frequency response to support the bound.Source: §III · p. 2
  • For a stable closed-loop system, stochastic LQR cost equals the squared H2 norm of its transfer system from process noise to the weighted state and control output.The paper uses the equivalence between H2 norm and steady-state mean-square output to establish this connection.Source: §III · pp. 2–3
  • The H2 perturbation lemma bounds the difference between two stable systems' H2 norms when their state matrices are sufficiently close.The proof bounds the transfer-function difference over |z| ≥ 1 using resolvents and then applies the H2-norm bound.Source: §III · p. 3
  • The proof controls transfer-function changes by separating output-matrix differences from resolvent differences, then applying triangle inequality and submultiplicativity.The resolvent-based argument requires ˆϵ < 0.5/ˆcA1.Source: §III · p. 3

IV. MAIN RESULTS

Section IV uses the connection between LQR cost and the H2 norm to derive a computable bound on DARE-solution differences under sufficiently small LQR-data perturbations. The bound is linear in perturbation size to first order, uses a frequency-domain stability measure, and starts from weaker detectability assumptions than prior bounds.

IV. MAIN RESULTS

  • Lemma 4.2 bounds λmax(P2−P1) using only Problem 1 data, including a closed-loop resolvent H∞-norm computable by standard numerical algorithms.Source: §IV · p. 4
  • Reversing the systems gives a corresponding bound on λmax(P1−P2) that depends only on Problem 2 data.Source: §IV · p. 4
  • For sufficiently small ε, Theorem 4.3 bounds ∥P2−P1∥2 when the A, B, and C matrices differ by at most ε.The bound is explicit and computable; an expression for the threshold ε0 in terms of Problem 1 data is omitted because it requires a homotopy argument.Source: §IV · pp. 3–5
  • The bound on λmax(P2−P1) is linear in ε to first order, with the remaining factors depending on Problem 1 data.Source: §IV · p. 5
  • Both approaches yield a bound linear in ε, but the bounds are not directly comparable because one uses a time-domain stability measure and the other a frequency-domain measure.Source: §IV · p. 6
  • Unlike the compared result’s observability assumption, this bound requires only detectability and permits positive semidefinite DARE solutions.This allows application in more general situations.Source: §IV · p. 6

V. CONCLUSION

The letter derives perturbation bounds for discrete Riccati equations by relating time- and frequency-domain properties. The bounds address systems close in matrix 2-norm, and future work may extend the approach to other control objectives with suitable frequency-domain equivalencies.

V. CONCLUSION

  • The approach uses time- and frequency-domain relationships to bound the norm of Riccati solutions for systems close in matrix 2-norm.Source: §V · p. 6
  • The letter derives novel perturbation bounds for the discrete Riccati equation.Source: §V · p. 6
  • Future work may extend the approach to other control objectives with suitable frequency-domain equivalencies.Source: §V · p. 6

APPENDIX A SUPPLEMENTAL LEMMAS

The appendix develops supporting bounds for differences in LQR gains and for resolvents of perturbed closed-loop matrices. These results use bounds on Riccati-related quantities and a sufficient closeness condition for preserving stability.

APPENDIX A SUPPLEMENTAL LEMMAS

  • The gain-difference proof bounds ΔS using ΔB and ΔP, then applies those bounds to bound ∥K1 − K2∥2.Source: Appendix A · p. 6
  • If ϵ < 0.5/ĉA1, the perturbed matrix has eigenvalues strictly inside the unit disk and its resolvent is well-defined for |z| ≥ 1.Under this condition, the resolvent norm is bounded by 2ĉA1.Source: Appendix A · p. 7
  • The resolvent identity and bounds on both resolvent norms yield a bound on the resolvent difference in terms of matrix closeness.Source: Appendix A · p. 7
Loading 2609.38515v1…