Kalman–Yakubovich–Popov lemma¶

A theorem equating a frequency-domain positivity condition for a linear system with existence of a state-space quadratic certificate.

Version: v1 · 2026-09-08 · History
Domain-specific #: 5173
Origin domain: control theory
Subdomain: control theory
Aliases: KYP lemma

Core Idea¶

Multiple strict nonstrict discrete-time and matrix-valued forms exist, controllability detectability and Hurwitz assumptions depend on the version and it is a mathematical equivalence rather than a controller-tuning recipe. A transfer-function inequality over all frequencies is converted into a Lyapunov or linear-matrix equation or inequality with a symmetric storage matrix, linking input-output positivity to dissipativity and stability in state space. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

Scope of Application¶

Kalman–Yakubovich–Popov lemma belongs to control theory and is useful where the analyst can specify the typed control theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the continuous or discrete linear time-invariant state-space matrices, stability and controllability or minimality assumptions, frequency-domain transfer expression and positivity inequality, symmetric matrix P and auxiliary factors, Lyapunov or LMI relation, strict or nonstrict version, storage function and dissipativity or positive-real interpretation and equivalence directions are explicit.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the continuous or discrete linear time-invariant state-space matrices, stability and controllability or minimality assumptions, frequency-domain transfer expression and positivity inequality, symmetric matrix P and auxiliary factors, Lyapunov or LMI relation, strict or nonstrict version, storage function and dissipativity or positive-real interpretation and equivalence directions are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Kalman–Yakubovich–Popov lemma. Kalman–Yakubovich–Popov lemma compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed control theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem. 2. Lock the constitutive rule. Express the continuous or discrete linear time-invariant state-space matrices, stability and controllability or minimality assumptions, frequency-domain transfer expression and positivity inequality, symmetric matrix P and auxiliary factors, Lyapunov or LMI relation, strict or nonstrict version, storage function and dissipativity or positive-real interpretation and equivalence directions are explicit independently of one notation or implementation.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of control theory because they reuse the typed control theory carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, A transfer-function inequality over all frequencies is converted into a Lyapunov or linear-matrix equation or inequality with a symmetric storage matrix, linking input-output positivity to dissipativity and stability in state space., and type the carrier, state every parameter and convention in the definition, test that the continuous or discrete linear time-invariant state-space matrices, stability and controllability or minimality assumptions, frequency-domain transfer expression and positivity inequality, symmetric matrix P and auxiliary factors, Lyapunov or LMI relation, strict or nonstrict version, storage function and dissipativity or positive-real interpretation and equivalence directions are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases.

Relationships to Other Abstractions¶

Current abstraction Kalman–Yakubovich–Popov lemma Domain-specific

Parents (1) — more general patterns this builds on

Kalman–Yakubovich–Popov lemma is a kind of Verification Prime

The proposed strict upward parent is prime:verification.

Hierarchy path (1) — routes to 1 parentless root

Kalman–Yakubovich–Popov lemma → Verification → Evaluation → Comparison → Self Checking

Neighborhood in Abstraction Space¶

Kalman–Yakubovich–Popov lemma sits in a crowded region of the domain-specific corpus (28^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Feedback Control & Dynamical Systems (29 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-09-08