Linear Parameter Varying (LPV) subspace identification methods suffer from an exponential growth in number of parameters to estimate. This results in problems with ill-conditioning. In literature, attempts have been made to address the ill-conditioning by using regularization. Its effectiveness hinges on suitable a priori knowledge. In this paper we propose using a novel, alternative regularization. That is, we first show that the LPV sub-Markov parameters can be organized into several tensors which are multi-linear low-rank by construction. Namely, their matricization along any mode is a low-rank matrix. Then we propose a novel convex method with tensor nuclear norm regularization which exploits this low-rank property. Simulation results show that the novel method can have higher performance than the regularized LPV-PBSIDopt technique in terms of variance accounted for.

In this paper, we present a novel multiple input multiple output (MIMO) linear parameter varying (LPV) state-space refinement system identification algorithm that uses tensor networks. Its novelty mainly lies in representing the LPV sub-Markov parameters, data and state-revealing matrix condensely and in exact manner using specific tensor networks. These representations circumvent the ‘curse-of-dimensionality’ as they inherit the properties of tensor trains. The proposed algorithm is ‘curse-of-dimensionality’-free in memory and computation and has conditioning guarantees. Its performance is illustrated using simulation cases and additionally compared with existing methods.

The major bottleneck in state-of-the-art Linear Parameter Varying (LPV) subspace methods is the curse-of-dimensionality during the first regression step. In this paper, the origin of the curse-of-dimensionality is pinpointed and subsequently a novel method is proposed which does not suffer from this bottleneck. The problem is related to the LPV sub-Markov parameters. These have inherent structure and are dependent on each other. But state-of-the-art LPV subspace methods parametrize the LPV sub-Markov parameters independently. This means the inherent structure is not preserved in the parametrization. In turn this leads to a superfluous parametrization with the curse-of-dimensionality. The solution lies in using parametrizations which preserve the inherent structure sufficiently to avoid the curse-of-dimensionality. In this paper a novel method based on tensor regression is proposed. This novel method is named the Predictor-Based Tensor Regression method (PBTR). This method preserves the inherent structure sufficiently to avoid the curse-of-dimensionality. Simulation results show that PBTR has superior performance with respect to both state-of-the-art LPV subspace techniques and also non-convex techniques.