Dual Youla Parametrization in LTI Control

• Dual Youla Parametrization is a framework that parametrizes all closed-loop stabilizable plant models via coprime factorization and affine constraints, enabling robust identification for LTI systems.
• Modern variants eliminate explicit coprime factorization by directly linking closed-loop data to convex controller synthesis and simplified predictor designs.
• The approach effectively addresses structural constraints and quadratic invariance, ensuring computational efficiency and accurate frequency response estimation even for unstable plants.

The Dual Youla Parametrization is a system-theoretic framework central to closed-loop control identification and synthesis for linear time-invariant (LTI) systems. It parametrizes all plant models that can be stabilized by a given controller using coprime factorization, yielding robust identification and synthesis methodologies even for unstable systems. Modern variants eliminate the need for explicit coprime factorization, enabling direct closed-loop identification and convex controller synthesis, while maintaining the theoretical rigor and practical robustness of the classical approach.

1. Classical Dual Youla Parametrization for Closed-Loop Identification

The dual Youla approach is formulated for closed-loop data where the true plant $P$ and stabilizing controller $C$ are interconnected:

$$\begin{cases} y = P\,u + w, \ u = r - C\,y + \xi, \end{cases}$$

with $r$ as external reference, $w$ as measurement noise, and $\xi$ as input disturbance. The approach leverages the doubly coprime factorization over $\mathcal{RH}_\infty$. For left-coprime factorizations,

$$C = D_K^{-1} N_K,\quad D_K, N_K \in \mathcal{RH}_\infty,$$

and a Bézout identity is satisfied along with $(D_0, N_0)$:

$D_0 \widehat{D}_K + N_0 \widehat{N}_K = I_p,$

where $\widehat{D}_K, \widehat{N}_K$ form a right-coprime factorization of $C$. The set of all plants stabilized by $C$ is parametrized by the stable transfer $Q \in \mathcal{RH}_\infty^{p \times p}$ as

$P = (D_0 - Q N_K)^{-1}(N_0 + Q D_K).$

By measuring internal signals $\alpha = D_K u + N_K y$ and $\beta_m = D_0 y - N_0 u$, the relation

$\beta_m = Q \alpha + \eta$

reduces closed-loop identification to an open-loop regression with noise $\eta$. The plant is identified by:

1. Estimating $Q$ via least squares:

$J(Q) = \|\beta_m - Q \alpha\|_2^2$

2. Recovering $P$ by inverting the dual Youla map for the estimated $Q$ (Sugie et al., 2020).

2. Algebraic Foundations and Affine Structure

The classical dual Youla parametrization and input-output parametrizations are built upon affine relationships among the closed-loop transfer maps. For the standard feedback interconnection with plant $G$ and controller $K$, the closed-loop transfer matrices are:

$$\begin{bmatrix} y \ u \end{bmatrix} = \begin{bmatrix} X & W \ Y & Z \end{bmatrix} \begin{bmatrix} w_y \ w_u \end{bmatrix}$$

with

$$\begin{aligned} X &= (I-GK)^{-1}, \ Y &= K(I-GK)^{-1}, \ W &= (I-GK)^{-1}G, \ Z &= (I-KG)^{-1}. \end{aligned}$$

These matrices satisfy the affine constraints: $$\begin{bmatrix} I & -G \end{bmatrix} \begin{bmatrix} X & W \ Y & Z \end{bmatrix} = \begin{bmatrix} I & 0 \end{bmatrix},$$

$$\begin{bmatrix} X & W \ Y & Z \end{bmatrix} \begin{bmatrix} -G \ I \end{bmatrix} = \begin{bmatrix} 0 \ I \end{bmatrix}$$

This affine subspace encodes all stabilizing controllers, with $K = Y X^{-1}$ (Furieri et al., 2019).

3. Simplified Dual Youla Approach and Stabilized PEM

The main practical limitation of the classical approach is the requirement for explicit coprime factorizations. As shown by Sugie & Maruta, this step can be entirely bypassed. By algebraically expressing $Q$ in terms of $P$:

$Q = (D_0 P - N_0) (N_K P + D_K)^{-1},$

and substituting into the identification criterion, the problem can be cast entirely in terms of $P$ without separately identifying $Q$. The resulting one-step-ahead predictor for $y$ is:

$$\hat{y}(t|\theta) = [I + \hat{P}(q,\theta) C(q)]^{-1} \hat{P}(q,\theta) C(q)\, r(t),$$

with the cost function

$$J(\theta) = \sum_{t=1}^N \|y(t) - \hat{y}(t|\theta)\|^2,$$

which is the stabilized prediction error method (PEM) (Sugie et al., 2020). This predictor is well-posed for any stabilizing $\hat{P}$, including the case where the true plant is unstable.

4. Controller Synthesis: Connections to Input-Output and Youla Parametrizations

The input-output parametrization of stabilizing controllers is structurally equivalent to the Youla parametrization, but expressed directly in terms of the closed-loop transfer maps $(X, Y, W, Z)$. If a doubly-coprime factorization is available:

$G = N_r M_r^{-1} = M_\ell^{-1} N_\ell,$

then the classical Youla controller formula,

$K = (V_r - M_r Q)(U_r - N_r Q)^{-1},$

is equivalently encoded by choosing

$$X = (U_r - N_r Q)M_\ell, \quad Y = (V_r - M_r Q)M_\ell,$$

and so forth, ensuring $(X, Y, W, Z)$ satisfy the affine constraints. Conversely, for any feasible $(X, Y, W, Z)$, the corresponding Youla parameter $Q$ can be recovered explicitly. The essential equivalence is established algebraically (Furieri et al., 2019).

A significant practical advantage is that the input-output approach never requires explicit computation of coprime factors $(N_r, M_r, U_r, V_r, \dots)$, only the imposition of linear affine relations.

5. Finite-Dimensional Realization and Algorithmic Steps

To numerically implement the dual Youla or input-output parametrization, a finite impulse response (FIR) approximation is employed. Each stable transfer is expanded into a basis (e.g., in discrete time: $X(z) = \sum_{i=0}^N X[i] z^{-i}$). The affine constraints are rewritten into a finite set of linear equations in the coefficients $\{X[i], Y[i], W[i], Z[i]\}$.

The finite-dimensional synthesis or identification reduces to a convex program (LP or QP for $\mathcal{H}_2$ or $\ell_1$ objectives). The algorithmic steps include:

1. Choose a parametric model structure for $\hat{P}(q,\theta)$.
2. Implement the stabilized predictor.
3. Formulate the convex norm-minimization (e.g., least-squares error, $\mathcal{H}_2$, or $\ell_1$ norm).
4. Apply LP/QP solvers for finite-dimensional approximation.
5. Recover the plant or controller from optimal parameter estimates (Sugie et al., 2020, Furieri et al., 2019).

6. Structural Constraints and Quadratic Invariance

Imposing structural constraints (sparsity, delay, locality) on the controller $K$ is systematically addressed via the concept of quadratic invariance (QI). For a linear subspace constraint $\mathcal{K} \subset \mathcal{R}_p^{m \times p}$, QI with respect to $G$ holds if:

$K G K \in \mathcal{K},\;\forall K \in \mathcal{K}.$

This is equivalent, in the input-output picture, to $Y \in \mathcal{K}$. Thus, convex programming over the affine set defined by $(X, Y, W, Z)$, subject to $Y \in \mathcal{K}$, yields the exact norm-optimal structured controller for QI subspaces. No pre-computation of coprime factors or detailed state-space models is required, and the synthesis is achieved by direct convex optimization (Furieri et al., 2019).

7. Theoretical Merits and Practical Evidence

The dual Youla parametrization and its modern simplifications share important theoretical guarantees:

• Noise-model independence: No explicit parametric modeling of the noise is required for closed-loop identification, ensuring consistent estimation under colored noise.
• Compatibility with unstable plants: The identification and synthesis remain valid for unstable $P$ due to the stabilizing effect of the predictor structure.
• Controller robustness: Identification is empirically robust to model mismatch in $C$; sensitivity to $C$ errors appears only at the second order.
• Computational efficiency: The elimination of coprime factorization greatly lowers computational barriers, particularly for large-scale and industrial systems (Sugie et al., 2020).

Simulation studies on benchmark systems confirm accurate frequency response estimation and reduced bias/variance compared to direct ARX/ARMAX identification, especially when the controller is fixed or its structure is known. The framework enables direct synthesis and identification of distributed or structured controllers as convex programs without requiring pre-stabilization or elaborate factorization procedures (Sugie et al., 2020, Furieri et al., 2019).