sidLTVdiscIO¶

Python equivalent: sid.ltv_disc_io

Identify discrete-time LTV system from partial observations.

result = sidLTVdiscIO(Y, U, H, 'Lambda', lambda)
result = sidLTVdiscIO(Y, U, H, 'Lambda', lambda, 'R', R)
result = sidLTVdiscIO(Y, U, H, 'Lambda', lambda, 'TrustRegion', 1)

Identifies time-varying system matrices A(k), B(k) and estimates state trajectories from input-output data when only partial state observations are available:

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

Uses the Output-COSMIC algorithm: alternating minimisation between a COSMIC step (dynamics estimation) and an RTS smoother (state estimation), initialised via LTI realization from the I/O transfer function (sidLTIfreqIO).

When H = I (full state observation), this reduces to the standard COSMIC algorithm (sidLTVdisc).

Inputs¶

Name	Description
`Y`	Output data, (N+1 x py), (N+1 x py x L), or cell array {L x 1} where Y{l} is (N_l+1 x py). Cell arrays allow variable-length trajectories.
`U`	Input data, (N x q), (N x q x L), or cell array {L x 1} where U{l} is (N_l x q). Must match Y format.
`H`	Observation matrix, (py x n). When rank(H) = n (including py >= n), states are recovered exactly via weighted least squares and no EM iterations are needed.

Name-value options¶

Name	Description
`'Lambda'`	Regularisation strength. Scalar or (N-1 x 1). Required.
`'R'`	Measurement noise covariance, (py x py) SPD. Default: eye(py).
`'MaxIter'`	Maximum alternating iterations. Default: 50.
`'Tolerance'`	Convergence tolerance on relative cost change. Default: 1e-6.
`'CovarianceMode'`	How to estimate noise covariance. Options: 'diagonal' - diagonal Sigma (default) 'full' - full n x n covariance 'isotropic' - scalar * I_n
`'TrustRegion'`	Trust-region parameter mu_0 in [0, 1], or 'off'. Default: 'off'.
`'TrustRegionTol'`	Minimum mu before final pass. Default: 1e-6.

Outputs¶

Name	Description
`result`	Struct with fields:
`.A`	(n x n x N) estimated dynamics matrices
`.B`	(n x q x N) estimated input matrices
`.X`	(N+1 x n x L) or cell {L x 1} estimated states
`.H`	(py x n) observation matrix (copy)
`.R`	(py x py) noise covariance used
`.Cost`	(n_iter x 1) cost J at each iteration
`.Iterations`	scalar, number of alternating iterations
`.Lambda`	(N-1 x 1) regularisation used
`.DataLength`	N
`.StateDim`	n
`.OutputDim`	py
`.InputDim`	q
`.NumTrajectories`	L
`.Algorithm`	'cosmic'
`.Method`	'sidLTVdiscIO'

Examples¶

% Basic usage
result = sidLTVdiscIO(Y, U, H, 'Lambda', 1e5);

% With known measurement noise
result = sidLTVdiscIO(Y, U, H, 'Lambda', 1e5, 'R', R_meas);

% With trust-region for difficult convergence
result = sidLTVdiscIO(Y, U, H, 'Lambda', 1e5, 'TrustRegion', 1);

% Multi-trajectory
result = sidLTVdiscIO(Y_3d, U_3d, H, 'Lambda', 1e5);

Algorithm¶

Output-COSMIC alternating minimisation: 1. LTI initialisation: estimate A0, B0 via Ho-Kalman realization of the I/O transfer function (sidLTIfreqIO) 2. State step: fix dynamics, RTS smoother for states 3. COSMIC step: fix states, solve for A(k), B(k) 4. Repeat 2-3 until convergence Complexity: O(T * (LNn^3 + N*(n+q)^3)) where T is iterations.

References¶

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

Specification¶

SPEC.md section 8.12 -- Output-COSMIC docs/cosmic_output.md -- Full derivation

Changelog¶

2026-04-06: Expose CovarianceMode option (was hardcoded 'diagonal').
2026-04-01: First version by Pedro Lourenço.