Weighting pattern¶

Represent a linear time-varying system's zero-state input–output action by a two-time kernel formed from output, state-transition, and input maps, reducing to an impulse-response convolution kernel in the time-invariant case.

Version: v2 · 2026-08-30 · History
Domain-specific #: 3105
Origin domain: linear systems theory
Subdomain: state space input output representations

Core Idea¶

The weighting pattern of a continuous-time linear state-space system is the two-time kernel (T(t,sigma)=C(t)Phi(t,sigma)B(sigma)) that maps past input at time (sigma) to its zero-state contribution to output at time (t).^[1] Variation of constants propagates an input injected through (B(sigma)) from (sigma) to (t) by (Phi(t,sigma)), then observes it through (C(t)); integrating (T(t,sigma)u(sigma)) over the causal interval yields the forced output separately from (C(t)Phi(t,t_0)x(t_0))

Its autonomous residual is the two-time input–output kernel of a linear dynamical system and its realization relation, not a cost-weight matrix, an arbitrary window function, or the state-transition matrix alone. The identity fails when the initial-state term is folded incorrectly into the kernel, matrix dimensions do not compose, the kernel is assumed to depend only on time difference in a time-varying system, causality limits are omitted, or one realization is identified uniquely from external behavior.

Recognition requires an analyst to state the state-space equations and dimensions, solve the homogeneous transition equation, separate initial-state and forced terms, derive the kernel, verify the integration or summation limits and causality, and test the time-invariant reduction. Once established, it supports comparing input–output-equivalent realizations, deriving impulse responses, separating state and external behavior, analyzing causality, and moving between time-varying state-space and kernel descriptions without turning those uses into the definition.

Structural Signature¶

Carrier: a linear time-varying state-space system (dot x=A(t)x+B(t)u), (y=C(t)x), with state-transition matrix (Phi(t,sigma))
Inputs or antecedent state: state, input and output spaces, coefficient matrices, initial time and state, state-transition matrix, admissible input, two time arguments, causality convention, and integral or sum convention
Constitutive operation: Variation of constants propagates an input injected through (B(sigma)) from (sigma) to (t) by (Phi(t,sigma)), then observes it through (C(t)); integrating (T(t,sigma)u(sigma)) over the causal interval yields the forced output separately from (C(t)Phi(t,t_0)x(t_0))
Invariant: the two-time kernel equals the composed input, transition, and output maps and reproduces the linear system's zero-state input–output response under the declared continuous or discrete convention
Recognition test: state the state-space equations and dimensions, solve the homogeneous transition equation, separate initial-state and forced terms, derive the kernel, verify the integration or summation limits and causality, and test the time-invariant reduction
Output or consequence: comparing input–output-equivalent realizations, deriving impulse responses, separating state and external behavior, analyzing causality, and moving between time-varying state-space and kernel descriptions
Failure boundary: the initial-state term is folded incorrectly into the kernel, matrix dimensions do not compose, the kernel is assumed to depend only on time difference in a time-varying system, causality limits are omitted, or one realization is identified uniquely from external behavior

What It Is Not¶

It is not the whole field of linear systems theory; many objects in that field do not satisfy its constitutive rule.
It is not its canonical example. For zero initial state, (y(t)=int_{t_0}^{t}T(t,sigma)u(sigma),dsigma) with (T(t,sigma)=C(t)Phi(t,sigma)B(sigma)). That is an instance, not a definition.
It is not State-Transition Matrix. The transition matrix propagates internal state under homogeneous dynamics; the weighting pattern composes that propagation with input injection and output observation to describe external forced response.
It is not an unrestricted metaphor. Direct feedthrough adds a distributional or instantaneous term involving D(t), while discrete time shifts the matrix power and summation indices according to the chosen update and output convention

Scope of Application¶

Weighting pattern applies when the analyst can specify a linear time-varying state-space system (dot x=A(t)x+B(t)u), (y=C(t)x), with state-transition matrix (Phi(t,sigma)) and establish that the two-time kernel equals the composed input, transition, and output maps and reproduces the linear system's zero-state input–output response under the declared continuous or discrete convention. The entry treats deterministic finite-dimensional linear systems under stated regularity assumptions. Stochastic kernels, nonlinear Volterra series, and optimization weights are separate objects.^[2]

Recognition. state the state-space equations and dimensions, solve the homogeneous transition equation, separate initial-state and forced terms, derive the kernel, verify the integration or summation limits and causality, and test the time-invariant reduction
Comparison. Compare legitimate instances through continuous or discrete time, time variance, initial state, direct feedthrough, causality, input and output dimensions, realization order, stability, integrability, and equivalence.
Boundary. Direct feedthrough adds a distributional or instantaneous term involving D(t), while discrete time shifts the matrix power and summation indices according to the chosen update and output convention
Use. Preserve every assumption when using the identity for comparing input–output-equivalent realizations, deriving impulse responses, separating state and external behavior, analyzing causality, and moving between time-varying state-space and kernel descriptions.

Clarity¶

A clear claim names the carrier, governing rule, assumptions, and recognition test. This matters because weighting pattern is older systems terminology and can be mistaken for statistical weights; the two-time causal kernel and realization role must be explicit. The disciplined statement is that the object counts as Weighting pattern exactly when the two-time kernel equals the composed input, transition, and output maps and reproduces the linear system's zero-state input–output response under the declared continuous or discrete convention

Identity and measurement remain separate. A kernel estimated from data has uncertainty and identifiability limits; equality of input–output behavior does not identify a unique internal realization without additional assumptions. Approximation or noisy evidence may weaken a classification without changing its definition.

Manages Complexity¶

The abstraction compresses continuous and discrete kernels, scalar and multivariable systems, time-invariant impulse responses, direct-feedthrough extensions, finite-horizon operators, and multiple state-space realizations into a stable carrier, rule, invariant, and failure boundary. It makes comparison tractable while retaining the variables that control validity.

Compression can hide assumptions. A responsible use therefore declares continuous or discrete time, time variance, initial state, direct feedthrough, causality, input and output dimensions, realization order, stability, integrability, and equivalence and returns to the full diagnostic whenever a convention or boundary case changes.

Abstract Reasoning¶

Type the carrier. Establish a linear time-varying state-space system (dot x=A(t)x+B(t)u), (y=C(t)x), with state-transition matrix (Phi(t,sigma)) and reject examples from a different problem.
Lock the rule. Express that the two-time kernel equals the composed input, transition, and output maps and reproduces the linear system's zero-state input–output response under the declared continuous or discrete convention independently of one notation or implementation.
Derive carefully. Infer comparing input–output-equivalent realizations, deriving impulse responses, separating state and external behavior, analyzing causality, and moving between time-varying state-space and kernel descriptions only under the stated assumptions.
Stress-test. Contrast the legitimate boundary case—Direct feedthrough adds a distributional or instantaneous term involving D(t), while discrete time shifts the matrix power and summation indices according to the chosen update and output convention—with this counterexample: an LQR matrix Q that weights state error in an optimization objective is not a weighting pattern because it does not encode the system's input–output kernel.

Knowledge Transfer¶

Transfer within linear systems theory is strong when new cases preserve the same carrier, mechanism, and diagnostic. The move from For zero initial state, (y(t)=int_{t_0}^{t}T(t,sigma)u(sigma),dsigma) with (T(t,sigma)=C(t)Phi(t,sigma)B(sigma)). to For a continuous-time LTI realization, (T(t,sigma)=Ce^{A(t-sigma)}B) for \(t\geqsigma\), so the zero-state response becomes convolution with the impulse response. demonstrates that continuity.^[3]

Outside the domain, only the skeleton—factor an external influence through injection, internal propagation, and observation, then aggregate its contributions over prior time—travels automatically. The terms state space, state-transition matrix, variation of constants, input-output map, kernel, impulse response, convolution, realization, causality, and feedthrough retain domain-specific meanings, so every role and inference must be revalidated.

Examples¶

Canonical¶

For zero initial state, (y(t)=int_{t_0}^{t}T(t,sigma)u(sigma),dsigma) with (T(t,sigma)=C(t)Phi(t,sigma)B(sigma)). The formula follows directly from variation of constants and keeps the initial-state response outside the input integral, correcting the common but dimensionally misleading substitution of (y(t_0)) for that term. It is canonical because the carrier, rule, invariant, and consequence are all inspectable.^[1]

Mapped back: a linear time-varying state-space system (dot x=A(t)x+B(t)u), (y=C(t)x), with state-transition matrix (Phi(t,sigma)) → Variation of constants propagates an input injected through (B(sigma)) from (sigma) to (t) by (Phi(t,sigma)), then observes it through (C(t)); integrating (T(t,sigma)u(sigma)) over the causal interval yields the forced output separately from (C(t)Phi(t,t_0)x(t_0)) → the two-time kernel equals the composed input, transition, and output maps and reproduces the linear system's zero-state input–output response under the declared continuous or discrete convention → comparing input–output-equivalent realizations, deriving impulse responses, separating state and external behavior, analyzing causality, and moving between time-varying state-space and kernel descriptions

Applied / In Practice¶

For a continuous-time LTI realization, (T(t,sigma)=Ce^{A(t-sigma)}B) for \(t\geqsigma\), so the zero-state response becomes convolution with the impulse response. Translation invariance collapses two time arguments to their difference; a time-varying system generally retains both arguments and is not represented by ordinary convolution. It qualifies only after the same diagnostic and failure boundary are checked.^[2]

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Exact identity vs. practical recognition. The constitutive condition may be exact while evidence is indirect. Diagnostic: Can the reviewer state both the condition and the warrant?
T2: Canonical form vs. variants. continuous and discrete kernels, scalar and multivariable systems, time-invariant impulse responses, direct-feedthrough extensions, finite-horizon operators, and multiple state-space realizations can preserve or change the identity. Diagnostic: Which named role is invariant across the variants?
T3: Compression vs. hidden assumptions. The label is useful only while prerequisites remain visible. Diagnostic: Can each downstream inference be traced to a declared assumption?
T4: Autonomy vs. reduction. The candidate uses broader structures but claims the two-time input–output kernel of a linear dynamical system and its realization relation, not a cost-weight matrix, an arbitrary window function, or the state-transition matrix alone. Diagnostic: Does that residual still support independent recognition after the parent and neighbors are subtracted?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is factor an external influence through injection, internal propagation, and observation, then aggregate its contributions over prior time; its identity-bearing terms are state space, state-transition matrix, variation of constants, input-output map, kernel, impulse response, convolution, realization, causality, and feedthrough. Those terms determine admissible objects, evidence, and consequences inside linear systems theory.

Structural Core vs. Domain Accent¶

The structural core is a carrier governed by Variation of constants propagates an input injected through (B(sigma)) from (sigma) to (t) by (Phi(t,sigma)), then observes it through (C(t)); integrating (T(t,sigma)u(sigma)) over the causal interval yields the forced output separately from (C(t)Phi(t,t_0)x(t_0)) and tested by state the state-space equations and dimensions, solve the homogeneous transition equation, separate initial-state and forced terms, derive the kernel, verify the integration or summation limits and causality, and test the time-invariant reduction. The domain accent is constitutive rather than decorative, so an analogy that preserves only the skeleton is not another instance of Weighting pattern.

The proposed strict upward parent is prime:representation. The kernel literally represents a linear system's external input–output behavior independently of a particular state realization; the two-time causal state-space factorization supplies the domain-specific residual. The edge is proposal-only and points to a frozen prior-baseline Prime.

The entry does not collapse into the parent because the two-time input–output kernel of a linear dynamical system and its realization relation, not a cost-weight matrix, an arbitrary window function, or the state-transition matrix alone A thematic neighbor is declined whenever it does not literally subsume that rule.

The prospective workspace queue contains one strict upward edge to prime:representation. No live DAG mutation is authorized.

Relationships to Other Abstractions¶

Current abstraction Weighting pattern Domain-specific

Parents (1) — more general patterns this builds on

Weighting pattern is a kind of Representation Prime

The proposed strict upward parent is prime:representation.

Hierarchy path (1) — routes to 1 parentless root

Weighting pattern → Representation → Abstraction

Neighborhood in Abstraction Space¶

Weighting pattern sits in a moderately populated region (59^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Feedback Control & Dynamical Systems (29 abstractions)

Nearest neighbors

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

Not to Be Confused With¶

Impulse response. The time-invariant single-lag specialization, often including feedthrough conventions.
State-transition matrix. Propagates state but does not by itself include B and C.
Transfer function. A transform-domain LTI representation that does not directly cover arbitrary time variation.
Weighting matrix. A matrix in a cost, covariance, or inner product rather than an input–output kernel.

References¶

[1] Roger W. Brockett, Finite Dimensional Linear Systems, Wiley, 1970, chapters 2–4, ISBN 978-0-471-10585-5. registry ↩a ↩b

[2] Wilson J. Rugh, Linear System Theory, 2^nd ed., Prentice Hall, 1996, chapters 3–6, ISBN 978-0-13-441205-4. registry ↩a ↩b

[3] Lotfi A. Zadeh and Charles A. Desoer, Linear System Theory: The State Space Approach, McGraw-Hill, 1963, sections on time-varying systems, state transition, and weighting patterns. registry ↩