LPV system identification using a separable least squares support vector machines approach

In this article, an algorithm to identify LPV State Space models for both continuous-time and discrete-time systems is proposed. The LPV state space system is in the Companion Reachable Canonical Form. The output vector coefficients are linear combinations of a set of a possibly infinite number of nonlinear basis functions dependent on the scheduling signal, the state matrix is either time invariant or a linear combination of a finite number of basis functions of the scheduling signal and the input vector is time invariant. This model structure, although simple, can describe accurately the behaviour of many nonlinear SISO systems by an adequate choice of the scheduling signal. It also partially solves the problems of structural bias caused by inaccurate selection of the basis functions and high variance of the estimates due to over-parameterisation. The use of an infinite number of basis functions in the output vector increases the flexibility to describe complex functions and makes it possible to learn the underlying dependencies of these coefficients from the data. A Least Squares Support Vector Machine (LS-SVM) approach is used to address the infinite dimension of the output coefficients. Since there is a linear dependence of the output on the output vector coefficients and, on the other hand, the LS-SVM solution is a nonlinear function of the state and input matrix coefficients, the LPV system is identified by minimising a quadratic function of the output function in a reduced parameter space; the minimisation of the error is performed by a separable approach where the parameters of the fixed matrices are calculated using a gradient method. The derivatives required by this algorithm are the output of either an LTI or an LPV (in the case of a time-varying SS matrix) system, that need to be simulated at every iteration. The effectiveness of the algorithm is assessed on several simulated examples.