Abstract-In this paper we present a Kalman-style realization theory for linear parameter-varying state-space representations whose matrices depend on the scheduling variables in an affine way (abbreviated as LPV-SSA representations). We show that such a LPV-SSA representation is a minimal (in the sense of having the least number of state-variables) representation of its input-output function, if and only if it is observable and span-reachable. We show that any two minimal LPV-SSA representations of the same input-output function are related by a linear isomorphism, and the isomorphism does not depend on the scheduling variable. We show that an input-output function can be represented by a LPV-SSA representation if and only if the Hankel-matrix of the input-output function has a finite rank. In fact, the rank of the Hankel-matrix gives the dimension of a minimal LPV-SSA representation. Moreover, we can formulate a counterpart of partial realization theory for LPV-SSA representation and prove correctness of the KalmanHo algorithm formulated in [1]. These results thus represent the basis of systems theory for LPV-SSA representation.

International audienceWhile determining the order as well as the matrices of a black-box linear state-space model is now an easy problem to solve, it is well-known that the estimated (fully parameterized) state-space matrices are unique modulo a non-singular similarity transformation matrix. This could have serious consequences if the system being identified is a real physical system. Indeed, if the true model contains physical parameters, then the identified system could no longer have the physical parameters in a form that can be extracted easily. By assuming that the system has been identified consistently in a fully parameterized form, the question addressed in this paper then is how to recover the physical parameters from this initially estimated black-box form. Two solutions to solve such a parameterization problem are more precisely introduced. First, a solution based on a null-space-based reformulation of a set of equations arising from the aforementioned similarity transformation problem is considered. Second, an algorithm dedicated to nonsmooth optimization is presented to transform the initial fully parameterized model into the structured state-space parameterization of the system to be identified. A specific constraint on the similarity transformation between both system representations is added to avoid singularity. By assuming that the physical state-space form is identifiable and the initial fully parameterized model is consistent, it is proved that the global solutions of these two optimization problems are unique. The proposed algorithms are presented, along with an example of a physical system

The problem of the online identification of multi-input multi-output (MIMO) state-space models in the framework of discrete-time subspace methods is considered in this paper. Several algorithms, based on a recursive formulation of the MIMO output error state-space (MOESP) identification class, are developed. The main goals of the proposed methods are to circumvent the huge complexity of eigenvalues or singular values decomposition techniques used by the offline algorithm and to provide consistent state-space matrices estimates in a noisy framework. The underlying principle consists in using the relationship between array signal processing and subspace identification to adjust the propagator method (originally developed in array signal processing) to track the subspace spanned by the observability matrix. The problem of the (coloured) disturbances acting on the system is solved by introducing an instrumental variable in the minimised cost functions. A particular attention is paid to the algorithmic development and to the computational cost. The benefits of these algorithms in comparison with existing methods are emphasised with a simulation study in time-invariant and time-varying scenarios. matrices.