An H∞-norm-based approach for operating point selection and LPV model identification from local experiments

Abstract

When the identification of linear parameter-varying (LPV) models from local experiments is considered, the question of the necessary number of local operating points as well as the problem of the efficient interpolation of the locally-estimated linear Keywords linear parameter-varying model · linear fractional representation · H ∞ -norm · non-smooth optimization · nu-gap metric IntroductionNowadays, the extensive demand for reliable models of nonlinear and/or time-varying systems requires the development of special model structures in which the non-linear behavior is broken down into several local models [16]. Among all the multi-model structures available in the literature, a particular attention has been paid to the linear parameter-varying (LPV) models during the last two decades (see [28, Chapter 1] for a historical review of LPV modeling and identification). This interest can be mainly explained by the following reasons. First, an LPV model can be seen as a combination of local models with parameters evolving as a function of measurable variables (called the scheduling variables) which can be related to the different operating points of the system. By this way, the model structure is close to the standard linear time-invariant (LTI) one but with a structural flexibility able to cope with time-varying, even non-linear behaviors. Second, the development of LPV models is linked to control engineering, where a controller must be designed in order to guarantee a suitable closed-loop performance for a given plant in different operating conditions. A well-known example of controller design technique using this "divide and conquer" basic idea is the gain scheduling approach [25]. A wide body of LPV controller design techniques is now available for this problem, which can be solved reliably, provided that a suitable model in a parameter-dependent form has been derived. LPV model identification methods based on I/O data [7]can be divided into two sub-classes generally called the global approach and the local approach, respectively. On the one hand, the global approach assumes that one global experiment can be performed during which the control inputs, as well as the scheduling variables, are both excited (see among others e.g., [15,5,32,14,31] and the references therein). By this way, all the nonlinearities of the system are excited at the same time by passing through a large number of operating points. On the other hand, local methods are based on a multi-step procedure where• local experiments are performed in which the operating points (corresponding to fixed values of the scheduling

Optimal identification experiment design for LPV systems using the local approach

“…Such an optimal LPV identification experiment design framework is novel. However, we had already tackled, in [13,23], the sub-problem of selecting optimal operating points for LPV system identification using the local approach. In [13], this problem is tackled supposing that each local LTI identification experiment yields a perfect model of the local LTI system (i.e.…”

“…With this assumption, the optimal operating points are defined as those leading to the best approximation of a complex parameter dependence on the scheduling function by a polynomial function of low order. In [23], an iterative methodology is developed to determine a set of operating points for which the corresponding local LTI systems represent a sufficient sample of the different local dynamics of the LPV system. However, unlike in the present paper, those operating points are not determined as the solution of an optimization problem and the variance of the identified local LTI models is also neglected.…”

Comparing global input-output behavior of frozen-equivalent LPV state-space models

“…By construction, these techniques are restricted to specific systems where the components of the scheduling variables are totally controllable and excitable (and not only measurable). On the other hand, the local approach is based on a multi-step procedure (see, e.g., Vizer and Mercère (2014), Vizer et al (2013b), Lovera and Mercère (2007)) where first, frozen linear time-invariant models are estimated for constant values of the the scheduling variables, then the global LPV model is built from the interpolation of the local LTI models. A substantial amount of local techniques are available in the literature (see De Caigny et al (2011), Luspay et al (2011)).…”

“…A substantial amount of local techniques are available in the literature (see De Caigny et al (2011), Luspay et al (2011)). This is probably due to the fact that the identification techniques for LTI systems are more mature than the LPV ones, in addition to the development of algorithms to optimize the number of local operating points involved in the procedure Vizer and Mercère (2014), as well as their strong link with the gain scheduling procedure Rugh and Shamma (2000). However, it is clear that none of these techniques is consistent when black-box state-space LPV models are estimated.…”

Combining Analytic and Experimental Information for Linear Parameter-Varying Model Identification: Application to a Flexible Robotic Manipulator

“…The second one, more experimental, consists of • the selection of a set of constant scheduling variables values for N op ∈ *  local working points so that all the working range of the system is covered and the "distance" between two constant working points is small enough, (see [16,39] for recent effective solutions for the determination of these local working points)…”