All-Stabilizing Youla-Kučera Architecture

• The paper shows that every internally stabilizing controller is uniquely generated by a stable free parameter Q using coprime or doubly-coprime factorizations.
• It establishes equivalences with alternative parameterizations, linking realization, system-level, input-output, and kernel approaches through affine mappings.
• Modern extensions incorporate convex reformulations, data-driven designs, and learning-based methods to enhance numerical efficiency and robust controller synthesis.

to=arxiv_search 菲律宾申博json {"query":"All-Stabilizing Youla-Kucera architecture Youla-Kucera parameterization realization-stability lemma", "max_results": 10} The all-stabilizing Youla–Kučera architecture is the classical controller parameterization in which a plant admitting a coprime or doubly-coprime factorization is paired with a stable free parameter $Q$, and every internally stabilizing controller is obtained from that parameter. In its standard form, the architecture separates stabilization from performance shaping: internal stability is enforced by the requirement $Q\in RH_\infty$, while closed-loop objectives are encoded through the choice of $Q$. Recent work places this architecture inside a broader family of realization-theoretic, kernel, system-level, input-output, operator-theoretic, data-driven, and neural constructions, but the core claim remains the same: the parameterization covers all stabilizing controllers for the specified plant class (Tseng, 2020).

1. Classical transfer-matrix form

For a right coprime factorization

$P(s)=N(s)M(s)^{-1},$

with $N,M\in RH_\infty$ proper, with no common unstable zeros, and with Bézout factors $X(s),Y(s)\in RH_\infty$ satisfying

$X(s)N(s)+Y(s)M(s)=I,$

the set of all internally stabilizing controllers can be written uniquely as

$$K(s)=\bigl(Y(s)-M(s)Q(s)\bigr)\bigl(X(s)+N(s)Q(s)\bigr)^{-1},$$

where $Q(s)\in RH_\infty$ is an arbitrary stable proper transfer matrix (Tseng, 2020). The “all-stabilizing” designation refers precisely to this one-to-one correspondence: every such $Q$ yields internal stability, and every internally stabilizing controller arises in this way.

Equivalent formulas appear under different coprime-factor conventions. With a doubly-coprime factorization

$Q\in RH_\infty$0

one obtains

$Q\in RH_\infty$1

and conversely every $Q\in RH_\infty$2 so plugged in yields a stabilizing $Q\in RH_\infty$3 (Zheng et al., 2019). The sign pattern and factor naming vary across sources, but the underlying content is the same: a stable free parameter spans the full internally stabilizing set.

The architecture is also a shaping mechanism. In the precision-motion formulation, with a nominal stabilizing controller $Q\in RH_\infty$4, loop gain $Q\in RH_\infty$5, and sensitivity $Q\in RH_\infty$6, introducing $Q\in RH_\infty$7 modifies the sensitivity as

$Q\in RH_\infty$8

so notches or peaks in $Q\in RH_\infty$9 can be placed by shaping $Q$0 (Hu et al., 19 Aug 2025). This makes explicit that the free parameter is not merely a proof device; it is a direct design coordinate for closed-loop frequency response.

2. Realization–stability viewpoint and equivalence with other parameterizations

A unifying interpretation is provided by the realization–stability lemma. If all internal signals are collected into a vector $Q$1, a realization matrix $Q$2 and an internal-stability matrix $Q$3 satisfy

$Q$4

Internal stability is then enforced by requiring every transfer from the external signal to each internal signal to lie in $Q$5, together with causality of the off-diagonal entries of $Q$6 (Tseng, 2020). In the Youla setting, substituting $Q$7 reduces the corresponding conditions to a doubly-coprime Bézout identity, from which the classical parameterization follows directly (Tseng, 2021).

This viewpoint matters because the Youla parameterization is not isolated. The realization–stability lemma shows that existing controller synthesis methods and realization proposals are all special cases of a simple lemma, and it enables easier equivalence proofs among existing methods (Tseng, 2020). In particular, explicit affine mappings link the Youla parameter $Q$8, the System-Level Parameterization (SLP), and the Input-Output Parameterization (IOP), so any convex problem in one coordinate system can be reformulated equivalently in the others (Zheng et al., 2019). A common misconception is that these are competing notions of stabilizing design; the cited work instead presents them as affine reparameterizations of the same stabilizing controller set.

The non-convexity of the controller set itself is preserved throughout. What becomes convex is the description in terms of closed-loop maps. One line of work identifies four groups of stable closed-loop transfer matrices equivalent to internal stability: one used in SLP, one used in IOP, and two new mixed forms, leading to four convex parameterizations of $Q$9 (Zheng et al., 2019). Under a fixed FIR horizon $P(s)=N(s)M(s)^{-1},$0, these parameterizations yield different inner approximations, and the IOP has the best ability of approximating $P(s)=N(s)M(s)^{-1},$1 given FIR constraints (Zheng et al., 2019). This suggests that the practical distinction among parameterizations is often computational and structural rather than fundamental.

3. Kernel and convex reformulations

A major modern development is the kernel version of the Youla parameterization. Focusing on left factors $P(s)=N(s)M(s)^{-1},$2, one obtains

$P(s)=N(s)M(s)^{-1},$3

Thus, instead of one free parameter $P(s)=N(s)M(s)^{-1},$4, all stabilizing controllers are described by two stable transfer matrices $P(s)=N(s)M(s)^{-1},$5 constrained by a single affine equation (Oliveira et al., 2022). This is still an all-stabilizing architecture, but it relocates the freedom from a fractional formula into an affine kernel relation.

The same paper shows that allowing the residual

$P(s)=N(s)M(s)^{-1},$6

and imposing

$P(s)=N(s)M(s)^{-1},$7

is equivalent to internal stability, by the small-gain theorem (Oliveira et al., 2022). In that form, stabilizing-controller synthesis becomes a right $P(s)=N(s)M(s)^{-1},$8-filtering problem,

$P(s)=N(s)M(s)^{-1},$9

This reinterpretation is technically significant because it imports finite-dimensional $N,M\in RH_\infty$0-LMI machinery into an otherwise infinite-dimensional parameterization.

Using a minimal state-space realization of $N,M\in RH_\infty$1 and a standard finite-dimensional $N,M\in RH_\infty$2-LMI lemma from Geromel–Bernussou or the extended version by de Scherer, the paper derives a single LMI of size $N,M\in RH_\infty$3, with $N,M\in RH_\infty$4, that is necessary and sufficient for $N,M\in RH_\infty$5 (Oliveira et al., 2022). Once $N,M\in RH_\infty$6 is obtained in state-space form with $N,M\in RH_\infty$7 invertible, the controller $N,M\in RH_\infty$8 is reconstructed explicitly and has internal order $N,M\in RH_\infty$9, matching the plant order (Oliveira et al., 2022). The paper states that this yields the first efficient Linear Matrix Inequality implicit parametrization of stabilizing controllers (Oliveira et al., 2022).

A different convex state-space variant appears in robust $X(s),Y(s)\in RH_\infty$0 guaranteed-cost control under parametric uncertainty. There, a variant of the Youla-Kučera parameterization is expressed by a symmetric matrix variable $X(s),Y(s)\in RH_\infty$1 and scalar $X(s),Y(s)\in RH_\infty$2, with the convex set

$X(s),Y(s)\in RH_\infty$3

and each feasible pair generates the stabilizing gain

$X(s),Y(s)\in RH_\infty$4

such that the closed loop is stable and $X(s),Y(s)\in RH_\infty$5 (Ma et al., 2020). In the uncertain case, the construction extends to all extreme vertices simultaneously, preserving convexity.

4. Numerical computation and robustness

The computational appeal of all-stabilizing parameterizations lies in the possibility of finite-dimensional convex synthesis, but numerical behavior differs sharply across formulations. For the kernel-LMI approach, the finite-dimensional problem is a single semidefinite program of size $X(s),Y(s)\in RH_\infty$6 returning an $X(s),Y(s)\in RH_\infty$7th-order controller, whereas FIR or Ritz approximations of Youla/SLP/IOP lead to SDPs of size $X(s),Y(s)\in RH_\infty$8 and controllers of order $X(s),Y(s)\in RH_\infty$9 (Oliveira et al., 2022). In chain-graph examples up to 14 subsystems, the LMI approach was an order of magnitude faster, in seconds rather than minutes, and produced 2nd-order local controllers, whereas a length-20 FIR SLP produced 1092nd-order controllers (Oliveira et al., 2022).

Numerical robustness is not uniform across parameterizations. For IOP, small mismatches in the affine constraints do not compromise closed-loop stability when the plant is open-loop stable; in that sense the IOP is numerically robust for open-loop stable plants (Zheng et al., 2019). The same work proves that a direct IOP implementation will fail to stabilize open-loop unstable systems in practice (Zheng et al., 2019). For SLP, numerical robustness of the two-block state-feedback controller is established, but numerical robustness of the four-block SLP controller requires case-by-case analysis even when the plant is open-loop stable (Zheng et al., 2019). A direct implication is that exact affine equivalence at the symbolic level does not imply equal floating-point behavior.

For large-scale robust programs, the state-space convex variant is paired with a symmetric Gauss–Seidel ADMM. The reformulated problem introduces consensus variables over the cone

$X(s)N(s)+Y(s)M(s)=I,$0

uses closed-form projection steps for the $X(s)N(s)+Y(s)M(s)=I,$1-update, a backward–forward symmetric Gauss–Seidel sweep for the $X(s)N(s)+Y(s)M(s)=I,$2-update, and a dual update with $X(s)N(s)+Y(s)M(s)=I,$3, with $X(s)N(s)+Y(s)M(s)=I,$4 used in the paper (Ma et al., 2020). The paper states that direct multi-block ADMM may fail to converge, whereas the symmetric Gauss–Seidel sweep recovers a two-block-like structure and yields linear convergence under the linear-quadratic non-smooth setting (Ma et al., 2020).

5. Data-driven and learning-based architectures

The all-stabilizing idea has also been recast in purely data-driven form. Using one long input-output trajectory $X(s)N(s)+Y(s)M(s)=I,$5 with $X(s)N(s)+Y(s)M(s)=I,$6 persistently exciting of order $X(s)N(s)+Y(s)M(s)=I,$7, one builds block-Hankel matrices $X(s)N(s)+Y(s)M(s)=I,$8 and $X(s)N(s)+Y(s)M(s)=I,$9. By Willems’ Fundamental Lemma, any length-$$K(s)=\bigl(Y(s)-M(s)Q(s)\bigr)\bigl(X(s)+N(s)Q(s)\bigr)^{-1},$$0 trajectory of the unknown LTI plant arises if and only if there exists $$K(s)=\bigl(Y(s)-M(s)Q(s)\bigr)\bigl(X(s)+N(s)Q(s)\bigr)^{-1},$$1 solving

$$K(s)=\bigl(Y(s)-M(s)Q(s)\bigr)\bigl(X(s)+N(s)Q(s)\bigr)^{-1},$$2

and, in the strictly proper case, the next output is predicted by $$K(s)=\bigl(Y(s)-M(s)Q(s)\bigr)\bigl(X(s)+N(s)Q(s)\bigr)^{-1},$$3 (Lawrence et al., 2023). This yields a data-driven internal model that can be inserted directly into a Youla realization: at each time step, the controller forms the internal Youla input $$K(s)=\bigl(Y(s)-M(s)Q(s)\bigr)\bigl(X(s)+N(s)Q(s)\bigr)^{-1},$$4, feeds it into a stable operator $$K(s)=\bigl(Y(s)-M(s)Q(s)\bigr)\bigl(X(s)+N(s)Q(s)\bigr)^{-1},$$5, and reproduces exactly the classical Youla–Kučera controller $$K(s)=\bigl(Y(s)-M(s)Q(s)\bigr)\bigl(X(s)+N(s)Q(s)\bigr)^{-1},$$6 for the unknown plant (Lawrence et al., 2023).

For linear $$K(s)=\bigl(Y(s)-M(s)Q(s)\bigr)\bigl(X(s)+N(s)Q(s)\bigr)^{-1},$$7, one paper gives an explicit unconstrained matrix factorization of all stable state matrices

$$K(s)=\bigl(Y(s)-M(s)Q(s)\bigr)\bigl(X(s)+N(s)Q(s)\bigr)^{-1},$$8

with $$K(s)=\bigl(Y(s)-M(s)Q(s)\bigr)\bigl(X(s)+N(s)Q(s)\bigr)^{-1},$$9 lower-triangular and $Q(s)\in RH_\infty$0 orthogonal, so that $Q(s)\in RH_\infty$1 is certified by a quadratic Lyapunov function (Lawrence et al., 2023). For nonlinear $Q(s)\in RH_\infty$2, a neural Lyapunov correction is used: $Q(s)\in RH_\infty$3 with $Q(s)\in RH_\infty$4 chosen in closed form to ensure $Q(s)\in RH_\infty$5 for $Q(s)\in RH_\infty$6, guaranteeing global exponential stability of the nonlinear dynamics (Lawrence et al., 2023). The same paper shows that if the measurement data are corrupted by i.i.d. Gaussian noise, then the induced operator $Q(s)\in RH_\infty$7 has spectral radius less than $Q(s)\in RH_\infty$8 with high probability, with $Q(s)\in RH_\infty$9 as $Q$0, so the open-loop data-driven predictor remains BIBO-stable despite noise (Lawrence et al., 2023).

These constructions have been integrated with reinforcement learning. In the two-tank study, the stable $Q$1-dynamics were trained with TD3, using a two-layer MLP for $Q$2, a two-layer input-convex network for $Q$3, and a linear readout $Q$4; 20 independent runs of 100 episodes each, at 0.5 s sampling and Gaussian measurement noise variance $Q$5, showed median cumulative reward converging after approximately 40 episodes, with no training runs exhibiting divergence (Lawrence et al., 2023). A related line based on recurrent equilibrium networks (RENs) proves that RENs are universal approximators of contracting and Lipschitz operators, and therefore that the Youla–REN architecture is a universal approximator of stabilizing nonlinear controllers for a partially-observed linear system (Wang et al., 2021). The central design principle is unchanged: optimization proceeds over a parameterization of stable $Q$6, so stability is guaranteed during learning transients (Wang et al., 2021).

6. Nonlinear, distributed, quantum, and application-specific extensions

In the operator-theoretic nonlinear setting, for a strictly-causal plant operator $Q$7, the all-stabilizing architecture takes the form

$Q$8

with closed-loop maps

$Q$9

Under these conditions, every $Q\in RH_\infty$00-stabilizing controller arises in this form for a unique $Q\in RH_\infty$01, and small-gain robustness to model mismatch is expressed by $Q\in RH_\infty$02 (Galimberti et al., 2024). This is an exact nonlinear analogue of the stable-parameter principle, though it assumes the nominal plant itself is $Q\in RH_\infty$03-stable.

For partially-observed nonlinear systems, the Youla architecture is built around a base stabilizing controller, a contracting and Lipschitz observer, the innovation $Q\in RH_\infty$04, and a free contracting and Lipschitz parameter $Q\in RH_\infty$05, often implemented as a REN (Barbara et al., 2 Jun 2025). The resulting augmented controller is

$Q\in RH_\infty$06

With any two among nonlinear dynamics, partial observation, and incremental closed-loop stability requirements, a contracting and Lipschitz Youla parameter leads to contracting and Lipschitz closed loops; if all three hold and exogenous disturbances are present, full incremental stability can be lost, and the preserved property is d-tube contraction and Lipschitzness (Barbara et al., 2 Jun 2025). The same work proves a disturbance-free converse and a partial converse under certainty-equivalence structure, which qualifies the meaning of “all-stabilizing” in the nonlinear disturbed case (Barbara et al., 2 Jun 2025). A local continuous-time state-space counterpart shows that every dynamic state-feedback controller that locally exponentially stabilizes a nonlinear input-affine system can be rewritten as a linear baseline $Q\in RH_\infty$07 plus the output of internal locally exponentially stable controller dynamics, yielding a local nonlinear Youla-type parametrization specialized to local exponential stability (Furieri, 5 Jan 2026).

Distributed and domain-specific versions further broaden the architecture. The Youla Operator State-Space framework characterizes stably realizable structured controllers over a network through stable operators $Q\in RH_\infty$08 and $Q\in RH_\infty$09, with

$Q\in RH_\infty$10

and under subspace-like assumptions the corresponding performance problem becomes a convex model-matching problem (Naghnaeian et al., 2019). A later distributed RL formulation embeds Graph Neural Networks into a Youla-like magnitude-direction parameterization, where a stable disturbance-feedback operator supplies the magnitude and a GNN on local observations supplies a unit-ball direction, guaranteeing network-level closed-loop stability by design (Cao et al., 20 Dec 2025). In sparse add-on design, all stabilizing system-level add-on controllers around a fixed decentralized baseline $Q\in RH_\infty$11 are parameterized by a static hollow Youla matrix $Q\in RH_\infty$12, giving

$Q\in RH_\infty$13

and an affine closed-loop map $Q\in RH_\infty$14 that yields a convex $Q\in RH_\infty$15 synthesis problem (Fan et al., 31 May 2026).

Application-specific loop shaping uses the same architecture for multi-band disturbance rejection. Starting from a baseline loop $Q\in RH_\infty$16 and a stable approximate inverse $Q\in RH_\infty$17, the augmenting controller becomes

$Q\in RH_\infty$18

and the sensitivity is approximately $Q\in RH_\infty$19 (Hu et al., 19 Aug 2025). Direct single-shot design becomes severely ill-conditioned beyond 4–5 notches, so the paper proposes an iterative multi-stage algorithm adding 1–2 notches at a time; in the hard-disk-drive case study, grouping frequencies into 6 groups of 2 each, with bandwidth around 30 Hz and stagewise order reduction to 2nd or 4th order, yields final 24th- or 48th-order controllers rather than a direct 240th-order design, with mean notch depth around 50 dB, peak high-frequency amplification of +6 dB max, and approximately 45% position-error-signal RMS reduction over baseline (Hu et al., 19 Aug 2025).

The architecture also extends to coherent quantum control. For a modified plant $Q\in RH_\infty$20 with loop block $Q\in RH_\infty$21, every internally stabilizing controller has the form

$Q\in RH_\infty$22

but physical realizability requires the controller to satisfy $Q\in RH_\infty$23-unitarity, which translates into the quadratic constraint

$Q\in RH_\infty$24

together with $Q\in RH_\infty$25 (Sichani et al., 2015). In that setting, the all-stabilizing parameterization remains intact, but admissible $Q\in RH_\infty$26 is restricted by quantum physical realizability.

Taken across these variants, the all-stabilizing Youla–Kučera architecture is best understood as a structural principle rather than a single formula. In classical LTI synthesis it is a stable transfer-matrix parameterization of all internally stabilizing controllers; in convex and kernel formulations it becomes an affine feasibility problem; in operator and neural settings it becomes a stable-operator search space; and in specialized domains it is adapted by additional constraints such as sparsity, network realizability, disturbance-feedback structure, or physical realizability. The common thread is the same one established by the classical theory: stabilization is encoded in the architecture, and performance optimization is carried out entirely within the stabilizing set.