State-Space Nonlinear Youla-Type Parametrization

• The paper details a state-space nonlinear Youla-type parametrization that decomposes dynamic controllers into a stabilizing linear feedback and a stable residual block to ensure local exponential stability.
• It establishes sufficiency and necessity of the parametrization via Lyapunov-based constructions and LMI conditions, linking classical linear Youla frameworks to nonlinear settings.
• Practical implications include extensions to operator-theoretic formulations, neural architectures, and data-driven synthesis for robust, distributed, and output-feedback control.

The state-space nonlinear Youla-type parametrization is a geometric and operator-theoretic framework that characterizes the complete set of dynamic state-feedback controllers attaining specified stability properties for nonlinear systems, often including guarantees of local or global exponential stability and robustness. This article surveys the theory and emerging practical synthesis algorithms for both continuous- and discrete-time nonlinear plants, operator-theoretic generalizations, neural augmentation, and learning-based controller design.

1. Mathematical Formulation for Input-Affine Nonlinear Systems

Consider a nonlinear, input-affine continuous-time plant: $$\dot x = f(x) + g(x)u,\qquad x\in\mathbb{R}^n,\ u\in\mathbb{R}^m$$ where $f$ and $g$ are $C^1$ maps, with $f(0)=0$ so that $x=0$ is an equilibrium for $u=0$. The system’s linearization at the origin is

$$\dot x = A x + B u,\quad A := \left.\frac{\partial f}{\partial x}\right|_{0},\quad B := g(0)$$

Assuming $(A,B)$ is stabilizable, i.e., there exists $K$ so that $A+B K$ is Hurwitz, the state-space nonlinear Youla-type parameterization provides a decomposition of all locally exponentially stabilizing dynamic state-feedback controllers as

$$\begin{cases} \dot z = A_z(z,x)\ u = K x + v(z,x) \end{cases}$$

Here $z \in \mathbb{R}^{n_z}$ is an internal controller state and $(A_z, v)$ are generally smooth, nonlinear maps whose linearization at the origin satisfies $A_z$ Hurwitz and $v$ affine in $(z,x)$. The key result is that every controller rendering $x=0$ locally exponentially stable admits such a representation for some $K$ and residual dynamics $(A_z, v)$ with $A_z$ Hurwitz (Furieri, 5 Jan 2026).

2. Equivalence and Stability Certificates

The main completeness theorem asserts:

• Sufficiency: Any controller of the given structure—combining a linear state-feedback $K x$ stabilizing the linearized plant with a residual dynamic block whose Jacobian is Hurwitz—yields local exponential closed-loop stability.
• Necessity: Any locally exponentially stabilizing dynamic state-feedback law for the equilibrium can, up to a smooth coordinate change, be put in this form; the linear gain $K$ may be taken as the Jacobian of the control law at the origin.

The proof leverages Lyapunov theory: constructing composite local Lyapunov functions for the plant state and controller state,

$V_{\mathrm{tot}}(x, z) = V_x(x) + \gamma V_z(z)$

with sufficiently large $\gamma$ to absorb all coupling terms, so that

$\dot V_{\mathrm{tot}} \le -\lambda \|[x, z]\|^2$

in a neighborhood of the origin (Furieri, 5 Jan 2026). Exponential stability reduces, in the linearized setting, to an LMI condition on the closed-loop matrix.

3. Connections to Classical Linear Youla Parameterization

In the LTI case ($f(x) = Ax$, $g(x) = B$), the nonlinear parameterization reduces to the standard Youla–Kučera framework, in which all stabilizing controllers have the transfer function

$K_\mathrm{full}(s) = K + Q(s)$

where $K$ is a fixed stabilizing gain and $Q(s)$ is any strictly proper, stable LTI map. The nonlinear framework replaces the transfer matrix $Q(s)$ with a state-space block of (possibly nonlinear) stable residual dynamics. Thus, the nonlinear Youla-type parametrization is a geometric lift of the classical affine parametrization (Furieri, 5 Jan 2026).

4. Operator-Theoretic and $\ell_p$-Stable Generalizations

For discrete-time nonlinear plants, an operator-theoretic version considers the plant as an operator $G : \ell^{n_u} \to \ell^{n_y}$, and parametrizes all $\ell_p$-stabilizing dynamic output-feedback controllers:

• Closed-loop responses $\Phi$ are fully characterized via a set of SLS-style operator constraints (Galimberti et al., 2024).
• Introducing a stable nonlinear operator $Q \in \mathcal{L}_p$, the controller internal map achieves closed-loop stability under small-gain or incremental finite-gain conditions, i.e., $\gamma_G \cdot \gamma_Q < 1$.
• The discrete-time IMC/Youla parameterization enables unconstrained, gradient-based optimization over all stabilizing $Q$.

The theory extends to robust distributed control via block-sparse $Q$, neural operator realization by Recurrent Equilibrium Networks (RENs), and data-driven identification by dual Youla parametrization (Galimberti et al., 2024, Boroujeni et al., 28 Mar 2025).

5. Neural Architectures and Learning Over Stabilizing Controllers

Neural implementations leverage contracting recurrent architectures, particularly RENs. The internal residual block is realized by a neural ODE or an equilibrium network with slope-restricted activations: $$\dot z = -\Lambda z + W \sigma(V[x;z]),\quad v = C_z z + D_z x$$ where $-\Lambda$ is Hurwitz and $\sigma$ is smooth. Training proceeds by backpropagation through time over closed-loop costs; internal stability is hard-constrained by design via structure and parameterization choices. As a result, all controller instantiations during learning are locally exponentially stable (Furieri, 5 Jan 2026, Wang et al., 2021, Wang et al., 2021). RENs are proven universal approximators of all contracting, Lipschitz Youla parameters subject to IQC bounds, admitting unconstrained learning by SGD.

6. Output-Feedback, Partial Observation, and Robustness Margins

Extensions include partial observation via observer-based architectures. The observer reconstructs the plant state; the Youla augmentation acts on innovations ($\tilde y_t = y_t - \hat y_t$): $$\hat x_{t+1} = f_o(\hat x_t, u_t, y_t),\quad u_t = k(\hat x_t) + Q(\tilde y_t)$$ Robustness is parametric: stability, contraction, and prescribed gain margins are enforced by REN architecture embedding incremental IQC bounds. For uncertain plants and disturbances, tube contraction properties quantify the robustness, and incremental gains can be tuned. Completeness of the parameterization holds for contracting and Lipschitz output-feedback controllers under modest additional assumptions (Barbara et al., 2 Jun 2025, Barbara et al., 2023).

7. Practical Control Synthesis and Algorithmic Implications

The state-space nonlinear Youla-type approach translates controller design for nonlinear plants into a convex (or unconstrained) search over the internal residual parameter $Q$:

1. Fix any stabilizing linear gain $K$ for the plant linearization.
2. Parameterize the residual dynamics by a stable (contracting, finite-gain) block—neural or otherwise.
3. Form the full controller $u = K x + Q(\cdot)$, where the block acts on plant state or innovation errors.
4. Train for performance via unconstrained optimization over $Q$, maintaining built-in guarantees of (local) exponential or robust stability.
5. For identification, use dual Youla techniques to recover closed-loop models stable under the observed controller (Boroujeni et al., 28 Mar 2025).

This geometric parametrization modularizes robust control design for nonlinear and uncertain plants, decoupling stability enforcement from performance optimization, and enabling safe learning-based controllers for output-feedback, distributed, and data-driven settings (Furieri, 5 Jan 2026, Galimberti et al., 2024, Wang et al., 2021, Wang et al., 2021, Barbara et al., 2 Jun 2025, Barbara et al., 2023, Boroujeni et al., 28 Mar 2025, Xu et al., 2023).