Minimal Volume Simplex (MVS) Polytopic Model Generation and Manipulation Methodology for TP Model Transformation

Abstract

The vertices of a polytopic LPV/qLPV model determine the feasibility of any linear matrix inequality (LMI) based control design as well as the achievable control performance, since the polytope typically includes not only the actual system dynamics. This paper proposes a powerful method to determine suitable polytopic qLPV Tensor‐Product model forms that allows for fine tuning through constraints on the locations of the vertices. The paper shows that these tuning methods are capable of effectively influencing the closed‐loop control performance results from LMI‐based multi‐objective optimisation. The advantages of the proposed methods are their tractability, low computational complexity, as well as their determinism. The key idea behind the proposed methods is to determine vertices in each decoupled parameter‐dimensions of the TP model in such a way that they form a minimal volume enclosing simplex fulfilling further fine tuning conditions on its shape. This is achieved based on the extension of the Minimal Volume Simplex Analysis (MVSA) method, where the volume of the enclosing simplex polytope is minimized by applying Sequential Quadratic Programming with majorization minimization strategy extended with a set of additional constraints influencing the location of certain vertices according to physically established considerations. Besides the comprehensive presentation of the proposed design methodology an expressive numerical example is provided to demonstrate its effectiveness and usability.

Affine Tensor Product Model Transformation

“…There exist numerical methods for enclosing simplex polytope generation (where = + 1) such as the Minimal 8 Complexity Volume Simplex Approach [13] and other simplex methods: CNO, IRNO, and SNNN [15,17]. The classical convex hull methods [28,29] can also be applied, but they usually result in enclosing polytopes with too many vertices (up to infinity).…”

“…Then the polytopic TP Model (13) can be formalized with w ( ) ( ) weighting functions and core tensor…”

“…Consolidating the affine geometry-based approach, the main contribution of this paper is the introduction of a new intermediate TP Model (like the HOSVD-based form) that provides a unique description of affine geometric properties serving as direct input for polytopic model construction methods (see [13,[15][16][17]). Furthermore, it reserves all the benefits of the HOSVD-based form: similar uniqueness, compact representation, and capability of complexity reduction.…”

“…A recent paper of the authors [13] established the affine geometric background of polytopic TP Model generation and proposed a direct way to determine the polytopic structures. First, it obtained the affine hulls of the subtensors of the discretized tensor and then the enclosing polytopes were established on the affine subspaces.…”

“…The paper proposes the Affine TP Model that substantially improves the polytopic TP Model generation and manipulation methodology by combining the affine geometric interpretations [13] with the benefits of higher-order SVD (HOSVD) based TP Model [2,12,14].…”