Operator-ADI balanced truncation of delay differential systems

1 Introduction

• Gramian approximation and balancing avoid discretizing the delay interval; the history component is represented exactly in the ADI and balancing steps.Takenaka–Malmquist coordinates represent the finite-rank ADI factors.Source: §1 · pp. 2–3
• Characteristic-root computations guide shift selection, and the procedure uses staged ADI shifts, goal-oriented stopping, and order selection.The roots are used only for choosing shifts, not for discretizing the delay during Gramian approximation or balancing.Source: §1 · pp. 2–3
• The method applies balanced truncation to the infinite-dimensional history-state realization and produces a finite-dimensional, delay-free reduced model.The reduced order is not tied to the instantaneous-state dimension n and need not be smaller than n.Source: §1 · pp. 2–3
• Under suitable conditions, the ADI-based reduced transfer functions converge in H∞ to those of exact infinite-dimensional balanced truncation.The paper directly verifies the reachability regularity needed for the convergence argument.Source: §1 · p. 3
• The construction extends to finitely many discrete retarded delays, while conjugate shift pairs use real 2n × 2n block systems.Real shifts require n × n systems, and the history components remain exactly represented.Source: §1 · p. 3
• The resulting models are compared with an independent Chebyshev discretization, and the complete MATLAB implementation used in the experiments is provided..Source: §1 · p. 3

2 Notation and preliminaries

• The paper fixes notation for bounded operators, adjoints, spectra, resolvents, and nuclear, Hilbert–Schmidt, spectral, and Frobenius norms.Source: §2 · p. 3
• The system framework uses a Hilbert space state, a generator of a strongly continuous semigroup, and bounded input and output operators.Source: §2 · p. 3
• The semigroup is assumed exponentially stable, and the stated assumptions are sufficient for the delay systems considered while avoiding additional technicalities.The paper notes that more general balanced-truncation and Lyapunov theory allows unbounded admissible control and observation operators.Source: §2 · pp. 3–4

3.1 Balanced truncation

• Balanced truncation produces a finite-dimensional delay-free system, with an error bound when the truncation does not split a multiple Hankel singular value.The factors of the reachability and observability Gramians provide a construction of the balanced realization.Source: §3.1 · pp. 4–5

3.2 ADI iteration and Takenaka–Malmquist systems

• The ADI factor is the reachability operator restricted to a Takenaka–Malmquist subspace, yielding finite-rank Gramian approximations.When the shifts satisfy the non-Blaschke condition, the system is complete in L2(R+), and the approximations converge to the reachability Gramian in nuclear norm.Source: §3.2 · pp. 5–6

3.3 ADI-based balanced truncation

• Replacing exact Gramian factors with ADI factors and applying square-root balancing gives an ADI-based reduced model.Its approximate Hankel singular values and leading singular vectors define the trial and test operators.Source: §3.3 · pp. 6–7
• Takenaka–Malmquist projections need not preserve the Hankel operator's shift structure, so their compressions are not generally Hankel operators of approximating systems.The analysis therefore works directly with the singular subspaces of the compressions.Source: §3.3 · p. 7
• Under a non-Blaschke shift condition and a gap σ_r > σ_{r+1}, the ADI-based reduced transfer functions converge in H∞ to exact balanced truncation.For the delay realization, the required reachability regularity is verified separately.Source: §3.3 · pp. 7–8

4 Delay systems

• The delay system has an infinite-dimensional state-space realization on C^n × L2(−h, 0; C^n), whose generator's spectrum is characterized by the zeros of det Δ(·).For this retarded system, exponential stability is characterized by the stated stability criterion.Source: §4 · pp. 8–9
• Exponential stability implies that the range of the reachability operator lies in the generator domain.The proof uses the fundamental matrix to establish the required regularity of reachable history states.Source: §4 · pp. 9–10
• For the delay realization, the convergence result holds when the shift sequences satisfy the non-Blaschke condition and σ_r > σ_{r+1}.The reachability-domain regularity required by the general theorem follows from exponential stability.Source: §4 · pp. 9–10

5 ADI on the delay state space

• Takenaka–Malmquist coordinates give exact finite-dimensional representations of all shifted resolvents used by operator ADI.When the basis is enlarged, prior history coefficients are retained by appending a zero block.Source: §5 · pp. 10–12

5.1 Realification of conjugate shift pairs

• A unitary blockwise change of basis turns conjugate-shift pairs into real-valued orthonormal functions spanning the same subspace.After shifts are grouped into real blocks, the implementation requires no complex arithmetic.Source: §5.1 · pp. 13–14

5.2 Real block ADI

• The real block iteration gives the same finite-rank Gramian approximations as complex operator ADI; real shifts use n × n solves and conjugate pairs use real 2n × 2n solves.The real block procedure represents the history components exactly.Source: §5.2 · pp. 14–15

5.3 Computational cost and storage

• After q scalar shifts, factor storage is organized in the shift dimension, and no discretization of the delay interval is stored.Required matrix-exponential updates are performed in q-dimensional shift space, with cost independent of the instantaneous-state dimension n.Source: §5.3 · p. 17

5.4 Shift parameter selection

• Shift selection refines rightmost characteristic-root candidates and reflects the retained roots across the imaginary axis to obtain ADI shifts.Successive bands of roots are processed from right to left; the search region can be enlarged if the available shifts are exhausted.Source: §5.4 · p. 18

6.1 Computation of the reduced model

• Real finite-dimensional ADI factors yield the balanced reduced model without delay-interval discretization or explicit infinite-dimensional balancing transformations.The approximate Hankel singular values are obtained from the factors and used in square-root balancing.Source: §6 · p. 18, §6.1 · pp. 19–20

6.2 Computational cost of the balancing step

• For small input and output dimensions, only applying A1 and cross products with X and Y involve the large state dimension n.The history contribution is computed using block exponentials for each input-output channel pair.Source: §6.2 · pp. 20–21

6.3 Stopping and order selection

• Order selection uses nested stable reduced models and frequency-grid diagnostics; iteration stops when the provisional order, stability, and both indicators meet prescribed conditions across consecutive checks.Orders are discarded if any of the three nested reduced models is unstable.Source: §6.3 · p. 21
• The indicators ηr and bεr are finite-factor diagnostics, not rigorous error bounds for the original delay system.Source: §6.3 · p. 22

• The multiple-delay extension assumes exponential stability, while the supplied implementation and numerical experiments use the single-delay formulation.Source: §7 · pp. 22–23
• For finitely many discrete delays, the characteristic-root refinement and shift construction remain unchanged except for replacing the single delay term with a finite sum.Source: §7 · p. 22
• The number of delays changes coefficient-matrix assembly but not shifted-system dimensions: real shifts require order n systems and conjugate pairs require real order 2n systems.Source: §7 · p. 23

8 Numerical experiments

• The experiments use real block ADI, whose cross products, approximate Hankel singular values, and reduced transfer functions agree to roundoff with complex Takenaka–Malmquist coordinates for identical shifts.The benchmark selects r = 19 although its instantaneous state dimension is n = 16, because reduction acts on the infinite-dimensional history-state space.Source: §8 · p. 23, §8.2 · p. 26

8.1 Validation and independent reference

• Against independent Chebyshev–Lobatto references, reduced models agree to approximately 10^-8 on Ωtest; the first 20 approximate Hankel singular values differ by less than 4 · 10^-5 from the N = 60 reference.The comparison uses the same order r = 15 for the operator-ADI model and the collocation balanced truncation.Source: Table 2 · p. 25, §8.1 · p. 25

8.2 A resonant mechanical delay benchmark

• The order-19 reduced model tracks the delay-system response across the plotted range, while removing the delayed term markedly changes resonances.The maximum sampled reduction error is approximately 0.227% of the delay system’s maximum sampled gain, attained at ω = 10^-3.Source: §8.2 · pp. 26–27, Figure 1 · p. 27
• The provisional order settles at r = 19 by J = 48; stabilization indicators pass at J = 80, and iteration stops at J = 96 after a second successful checkpoint.The global reachability residual remains several orders larger than the reduced-model change while the reduced input–output model has stabilized.Source: Table 3 · p. 28, §8.2 · p. 28

9 Conclusions

• Operator-ADI balanced truncation acts directly on the infinite-dimensional history-state realization while reducing the required computations to finite-dimensional linear algebra.Under suitable assumptions, the resulting reduced transfer functions converge in H∞ to exact infinite-dimensional balanced truncation.Source: §9 · p. 28
• The construction extends to finitely many discrete delays without increasing shifted linear-system dimensions; adaptive shift strategies and distributed or neutral delays remain future work.Source: §9 · p. 28
• The MATLAB implementation used for the numerical experiments is provided as supplementary material.Source: §9 · p. 28