Real plants are, in general, time-varying and uncertain. Yet most industrial control loops are designed using linear time-invariant (LTI) plant models. The present work formalizes the problem of best approximate LTI modelling of BIBO-stable linear time-varying (LTV) systems in a way that is compatible with the notion of induced l(infinity) norm used in robust l(1) control. This setup is closely associated with certain identification methods and control-motivated model validation/invalidation procedures that can be efficiently implemented with special-purpose linear programming (LP) techniques. Results are given for approximate modelling of BIBO-stable LTV and LTI systems using LTI models, especially BIBO-stable fixed-pole state-space model parametrizations. It is shown that such parametrizations are satisfactory from an approximate modelling point of view, and can be used in output-error-type LP identification techniques and in LP model validation procedures. Various useful constraints on model parameters etc. can be included, as long as they are linear in the unknown parameters, and both insensitive (statistically robust) and sensitive criteria can be used in identification and model validation. Simulation examples are included to illustrate that such techniques indeed give good results and can be used to solve problems of realistic size.