sidLTVdisc¶

Python equivalent: sid.ltv_disc

Identify discrete-time LTV state-space model.

result = sidLTVdisc(X, U)
result = sidLTVdisc(X, U, 'Lambda', lambda)
result = sidLTVdisc(X, U, 'Lambda', 'auto')
result = sidLTVdisc(X, U, 'Lambda', lambda, 'Uncertainty', true)
result = sidLTVdisc(X, U, ..., 'NoiseCov', Sigma)

Identifies time-varying system matrices A(k), B(k) from state and input trajectory data for the discrete LTV system:

\(x(k+1)\) = A(k) x(k) + B(k) u(k) k = 0, ..., N-1

Uses the COSMIC algorithm (Carvalho et al., 2022), which solves a regularized least squares problem balancing data fidelity against temporal smoothness of the system matrices via a closed-form block tridiagonal solver with O(N(p+q)^3) complexity.

Inputs¶

Name	Description
`X`	State data, one of: (N+1 x p) — single trajectory (N+1 x p x L) — L trajectories, same horizon {X1, X2, ...} — cell array of L trajectories with variable horizons (N_l+1 x p each)
`U`	Input data, matching format: (N x q) — single trajectory (N x q x L) — L trajectories, same horizon {U1, U2, ...} — cell array, (N_l x q) each

Name-value options¶

Name	Description
`'Lambda'`	Regularization strength. Options: scalar — uniform lambda at all time steps (N-1x1) — per-step lambda vector 'auto' — automatic selection via L-curve Default: 'auto'.
`'LambdaGrid'`	Candidate lambda values for L-curve auto selection. Only used when Lambda is 'auto'. Vector of positive scalars. Default: logspace(-3, 15, 50).
`'Precondition'`	Apply block-diagonal preconditioning to improve numerical stability. Default: false.
`'Algorithm'`	Identification algorithm. Currently only 'cosmic' is supported. Default: 'cosmic'.
`'Uncertainty'`	Compute Bayesian posterior uncertainty for A(k), B(k). Doubles computation cost. Default: false.
`'NoiseCov'`	Measurement noise covariance. Options: (p x p) — known noise covariance matrix 'estimate' — estimate from residuals (default) Requires 'Uncertainty' to be true (set automatically).
`'CovarianceMode'`	How to estimate noise covariance (when not user-provided). Options: 'diagonal' — diagonal Sigma (default) 'full' — full p x p covariance 'isotropic' — scalar * I_p Ignored when 'NoiseCov' is a matrix.

Outputs¶

Name	Description
`result`	Struct with fields:
`.A`	(p x p x N) time-varying dynamics matrices
`.B`	(p x q x N) time-varying input matrices
`.Lambda`	regularization values used, (N-1 x 1)
`.Cost`	[total, data_fidelity, regularization]
`.DataLength`	N (number of time steps)
`.StateDim`	p
`.InputDim`	q
`.NumTrajectories`	L
`.Algorithm`	'cosmic'
`.Preconditioned`	logical
`.Method`	'sidLTVdisc' When 'Uncertainty' is true, these fields are added:
`.AStd`	(p x p x N) std dev of A(k) entries
`.BStd`	(p x q x N) std dev of B(k) entries
`.P`	(d x d x N) row covariance, d = p+q
`.NoiseCov`	(p x p) noise covariance (provided or estimated)
`.NoiseCovEstimated`	logical, true if estimated from residuals
`.NoiseVariance`	scalar, trace(NoiseCov)/p
`.DegreesOfFreedom`	effective d.o.f. (NaN if NoiseCov provided)

Examples¶

% Basic identification with automatic lambda
result = sidLTVdisc(X, U);

% Manual uniform lambda
result = sidLTVdisc(X, U, 'Lambda', 1e5);

% With uncertainty estimation
result = sidLTVdisc(X, U, 'Lambda', 1e5, 'Uncertainty', true);

% With known noise covariance
Sigma = diag([0.01, 0.05]);  % known from sensor specs
result = sidLTVdisc(X, U, 'Lambda', 1e5, 'NoiseCov', Sigma);

% Per-step lambda (lower near a known transient at step 50)
lam = 1e5 * ones(N-1, 1);
lam(48:52) = 1e2;
result = sidLTVdisc(X, U, 'Lambda', lam);

Algorithm¶

COSMIC: Closed-form solution via block tridiagonal LU decomposition. Forward pass computes Lbd_k and Y_k; backward pass recovers C(k). Uncertainty: backward recursion on P(k) = [\(A^{-1}\)]_{kk} reusing stored Lbd_k, then Cov(vec(C(k))) = Sigma \otimes P(k). Complexity: O(N * (p+q)^3), linear in time steps.

References¶

Carvalho, Soares, Lourenco, Ventura. "COSMIC: fast closed-form identification from large-scale data for LTV systems." arXiv:2112.04355, 2022.

Specification¶

SPEC.md §8 — Discrete-Time LTV State-Space Identification

Changelog¶

2026-03-29: First version by Pedro Lourenço.
2026-04-01: Refactored into modular internal functions.