Master SDP/LMI Parameterizations

• Master SDP/LMI parameterizations are systematic representations that describe all feasible controllers, invariant sets, and system behaviors using affine LMI constraints.
• They enable scalable convex optimization techniques for robust controller synthesis and polynomial programming via lifted LMIs and hierarchical approximations.
• The framework supports data-driven analysis and uncertainty handling through efficient SDP formulations, exploiting decomposition methods and relaxation hierarchies.

A master parameterization for semidefinite programming (SDP) or linear matrix inequality (LMI) constraints refers to a systematic description of all feasible objects—controllers, invariant sets, Lyapunov functions, or system behaviors—using decision variables that enter the LMI constraints linearly or affinely, and which is sufficiently expressive to capture the entire solution set of interest or a large, efficiently optimizable subset. Such parameterizations are pivotal across systems and control theory, robust and data-driven optimization, verification, and polynomial programming, where convexity and scalability are decisive.

1. Foundational Concepts: Spectrahedra, Affine and Lifted LMIs

A spectrahedron is the basic convex set parameterized as the solution set of a linear matrix inequality: $$S = \{ x\in\mathbb{R}^n : F(x) := A_0 + \sum_{i=1}^n x_i A_i \succeq 0 \}$$ with symmetric pencils $A_i$. This is the natural SDP analogue to a polyhedron in linear programming. If all $A_i$ are diagonal, the spectrahedron reduces to a polyhedron; otherwise, the LMI encodes arbitrary coupling through the positive semidefinite (PSD) constraint (Henrion, 2013).

Not every convex basic semialgebraic set is describable as a direct spectrahedron. More generally, a set $S$ is semidefinite-representable (a spectrahedral shadow) if

$$S = \{ x : \exists y,\,\,A_0 + \sum x_i A_i + \sum y_j B_j \succeq 0 \}$$

where $y$ are auxiliary “lifting” variables. The geometry of such projected LMI-parameterized sets (“lifted LMIs”) is central in polynomial optimization, control, and verification (Henrion, 2013).

2. Hierarchical LMI Parameterizations: Stable Polynomial Sets

For root localization and controller synthesis, the feasible set is often nonconvex: e.g., for a real monic polynomial $p(z) = a_0 + a_1 z + \dots + z^n$, the set $S$ where $p$ is Schur-stable (all roots in $|z|<1$) has a complicated boundary. A hierarchy of inner LMI approximations can be constructed using spectral properties of Toeplitz matrices. For fixed $m>n$,

$S_m = \{ d \in \mathbb{R}^n : P_m^{c,d} \succ 0 \}$

with $P_m^{c,d}$ a Toeplitz pencil, forms a convex LMI-defined subset of the stability region. As $m\to\infty$, these convex sets converge to the maximal “lifted LMI” region, which itself corresponds to a projection of a $(n+1)\times(n+1)$ matrix pencil involving auxiliary variables (Rami et al., 2010). Each $S_m$ is an affine section of the PSD cone, with explicit, scalable LMI representations. This approach provides both theoretical hierarchy and practical tuning for fixed-order controller design via SDP.

3. Implicit and Explicit Controller Parameterizations

A primary application is the parameterization of all (internally) stabilizing controllers for a plant within an LMI-feasible set. Classical approaches (Youla–Kučera parameterization) describe all stabilizing controllers via an affine constraint in the frequency domain, but practical computation requires efficient finite-dimensional LMIs. The “kernel Youla” parameterization expresses all stabilizing controllers as: $$\mathcal{C}_{\mathrm{stab}} = \left\{ K = Y X^{-1} \mid M_\ell X - N_\ell Y = I,\, X, Y \in \mathcal{RH}_\infty \right\}$$ For discrete-time/doubly-coprime factorization. Introducing auxiliary stable transfer matrices ($X,Y$), the constraint $M_\ell X - N_\ell Y = I$ is affine and admits an efficient LMI characterization using robust $\mathcal{H}_\infty$ filtering machinery. The entire feasible set can then be computed via a single, finite LMI; this parameterization guarantees convexity, internal stability, and efficiently reconstructs a full-order controller of minimal order (Oliveira et al., 2022).

In the static output feedback (SOF) case, an LMI parameterization is achieved by freezing the Riccati solution $P$ from LQR and seeking $F$ that renders $A+BFC$ stable: $$\begin{pmatrix} Q - PBF C - C^T F^T B^T P & PB \ B^T P & R \end{pmatrix} > 0$$ This LMI is affine in $F$, with $P, Q, R$ set by Lyapunov and weighting requirements. The LQR gain is recovered as a special case, and similar SOF–LMI constraints extend to $\mathcal{H}_\infty$ and $\mathcal{H}_2$ performance objectives (Rodrigues, 2022).

4. Parameter-Dependent, Polynomial, and Set-theoretic LMIs

When the feasible set depends on additional (scheduling or design) parameters, master parameterizations extend naturally, although at significant computational and conceptual cost. For LPV systems, parameter-dependent Lyapunov functions and invariant sets are encoded by families of LMIs:

• For time-varying convex combinations $A(\theta) = \sum_{i=1}^r \theta_i A_i$, parameter-dependent Lyapunov functions $P(\theta) = \sum_{i=1}^r \theta_i P_i$ result in LMIs of the form $$A_i^T P_i + P_i A_i + \sum_{k=1}^r H_{k,j} P_k \prec 0$$. Combinatorial explosion is addressed through outer-bounding manipulations (simplex approximation), yielding master LMIs that scale linearly with the number of parameters (Mozelli et al., 2018).
• Robust invariant sets for LPV systems with parameter variation and bounded rate are parameterized as families of configuration-constrained polytopes:

$$\mathcal{S}(p | y^0, Y) = \{ x : Cx \leq y^0 + Yp \}$$

with concurrent parameterized vertex control laws and all set-inclusion and invariance conditions enforced by a single convex SDP using Farkas duality and the S-procedure (Mulagaleti et al., 2023).

• Parameter-dependent LMIs for invariant set computation with polynomial $P(\xi)$ and $K(\xi)$ are relaxed using Pólya’s theorem: LMIs are required only at the vertices of the simplex, yielding master parameterizations which are finite, explicit, and highly scalable, enabling both invariance and performance objectives (Gupta et al., 2020).

5. Master LMIs for Uncertainty, Data-Driven, and Robust Settings

Parameterizations extend beyond deterministic systems. For robust or data-driven settings:

• Distributionally robust synthesis (e.g., under Wasserstein ambiguity) is formulated as a master LMI via a Scherer-type congruence parameterization, introducing auxiliary variables $(X,Y,K,L,M,N)$ and block matrices that encode the effect of all possible probability distributions within a specified ball. The final controller synthesis is a convex SDP whose solution gives a robustly optimal controller (Gramlich et al., 27 Sep 2025).
• For data-driven system analysis under error-in-variables, the Sherman–Morrison–Woodbury identity yields a master LFT (linear fractional transformation) that encapsulates all systems consistent with the observed data and bounded uncertainties. Standard robust-control LMI machinery (full-block S-procedure multipliers) produces robust $\mathcal{H}_2$ or $\mathcal{H}_\infty$ performance guarantees with LMIs independent of sample count (Brändle et al., 2024).
• In data-driven pole-placement, Willems’ fundamental lemma is used to replace unknown system matrices with data matrices appearing linearly in the LMI constraints. The resulting SDP is equivalent to the classical controller design objective, provided persistency-of-excitation holds in the data (Mukherjee et al., 2021).

6. Algorithmic and Computational Considerations

Master parameterizations are justifiable only if LMI-based feasibility or optimization is tractable:

• The underlying convexity, block structure, and parameter sparsity permit decomposition techniques (e.g., chordal sparsity exploitation), turning large monolithic LMIs into equivalent collections of smaller ones without loss of feasibility or conservatism (Xue et al., 2022).
• For parametric LMIs (LMIs whose data depend polynomially on auxiliary parameters), semi-algebraic descriptions of feasible regions are constructed using critical-point and geometric resolution methods together with Hermite matrix reduction, enabling quantifier elimination and feasibility certification on Zariski-dense subsets (Naldi et al., 3 Mar 2025).
• Hierarchical relaxations (e.g., moment/SOS hierarchies) ensure theoretical convergence to the solution set, with computational costs and relaxation orders traded off for fidelity (Henrion, 2013, Streif et al., 2013, Claeys et al., 2011).

Table: Key Types of Master SDP/LMI Parameterizations

Domain	Master Parameterization	Reference
Stable polynomials	Hierarchy of Toeplitz-matrix LMIs $P_m^{c,d} \succ 0$	(Rami et al., 2010)
Controller design	Kernel Youla or affine static output feedback LMI	(Oliveira et al., 2022, Rodrigues, 2022)
Parametric/LPV	Affine/conic LMI with parameter-dependent Lyapunov or invariant set	(Mozelli et al., 2018, Mulagaleti et al., 2023, Gupta et al., 2020)
Robust/data-driven	LFT/Sherman–Morrison–Woodbury, distributionally robust block LMIs	(Gramlich et al., 27 Sep 2025, Brändle et al., 2024)
Verification/SOS	Hierarchies of moment/localizing matrices, spectrahedral shadows	(Henrion, 2013, Claeys et al., 2011, Streif et al., 2013)

7. Impact and Future Directions

Master parameterizations unify a diversity of control, optimization, and systems theory problems under a framework that can leverage convex optimization (SDP) technology. They are the basis for modern scalable algorithms in controller synthesis, system identification, formal verification of neural networks, polynomial optimization, and model validation. Open frontiers include minimal-lifting constructions for spectrahedral shadows, model-free and data-driven controller synthesis with robust performance criteria, scalable distributed SDP solvers exploiting problem structure, and universal parameterizations for noncommutative problems. Improvements in relaxation tightness, computational complexity control, and projection from lifted to explicit variable spaces remain crucial ongoing research directions (Henrion, 2013, Oliveira et al., 2022, Mulagaleti et al., 2023, Naldi et al., 3 Mar 2025).