By Péter Baranyi
Tensor Product version Transformation in Polytopic Model-Based keep watch over deals a brand new standpoint of keep an eye on process layout. rather than depending completely at the formula of more beneficial LMIs, that is the generally followed method in latest LMI-related reviews, this state of the art publication demands a scientific amendment and reshaping of the polytopic convex hull to accomplish superior functionality. various the convexity of the ensuing TP canonical shape is a key new characteristic of the method. The booklet concentrates on decreasing analytical derivations within the layout strategy, echoing the new paradigm shift at the reputation of numerical resolution as a legitimate type of output to regulate procedure difficulties. The salient good points of the booklet include:
provides a brand new HOSVD-based canonical illustration for (qLPV) types that allows trade-offs among approximation accuracy and computation complexity
helps a conceptually new keep watch over layout method through presenting TP version transformation that gives an easy approach of manipulating sorts of convexity to seem in polytopic representation
Introduces a numerical transformation that has the good thing about simply accommodating types defined by way of non-conventional modeling and identity techniques, reminiscent of neural networks and fuzzy rules
provides a couple of useful examples to illustrate the appliance of the method of generate keep watch over process layout for advanced (qLPV) platforms and a number of keep watch over pursuits.
The authors’ strategy is predicated on a longer model of singular price decomposition appropriate to hyperdimensional tensors. below the method, trade-offs among approximation accuracy and computation complexity could be played during the singular values to be retained within the strategy. using LMIs permits the incorporation of a number of functionality targets into the regulate layout challenge and coverage of an answer through convex optimization if possible. Tensor Product version Transformation in Polytopic Model-Based regulate contains examples and accommodates MATLAB® Toolbox TPtool. It presents a reference advisor for graduate scholars, researchers, engineers, and practitioners who're facing nonlinear platforms keep watch over purposes.
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Additional info for Tensor Product Model Transformation in Polytopic Model-Based Control
8) for the given qLPV model using only a minimal number of components. In the nonexact case, a trade-oﬀ study between the number of components and the accuracy of the resulting TP model is also provided. The procedures of the TP model transformation involve the discretization of the given qLPV model, and then using higher-order singular value decomposition (HOSVD) to obtain the unique tensor product structure of the given model. Based on this tensor structure we can readily identify the various components of the resultant TP model.
IN × (m + k) × (m + l)) and that of Un as (Mn × In ). If we are executing CHOSVD, then In = Rn = rankn (SD ). If we are executing RHOSVD, then In < Rn at least for one n. Hence, normally we have In ≤ Mn for all n = 1, . . , N. ,iN , in = 1, . . , In as elements. One may further deﬁne matrix transformations Tn to transform singular matrices ¯ n: Un to U ¯ n Tn = Un . 16) where S¯ = S N n=1 Tn . 17) ¯ n certain If transformation Tn is deﬁned in a special way, we can incorporate into U convexity types, and hence desirable convex hull, for the later control system design process.
N. ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ CHAPTER 2. 2. Computationally, the n-mode product of a tensor by a matrix, A = B ×n U, can be obtained by ﬁrst ﬁnding the n-mode matrix of B, B(n) , computing the product A(n) = UB(n) , and then converting A(n) to recover A. 1. Given the tensor A ∈ RI1 ×I2 ×···×IN and the matrices F ∈ R Jn ×In , G ∈ R Jm ×Im , n m, we have (A ×n F) ×m G = (A ×m G) ×n F = A ×n F ×m G. 2. Given the tensor A ∈ RI1 ×I2 ×···×IN and the matrices F ∈ R Jn ×In , G ∈ RKn ×Jn , we have (A ×n F) ×n G = A ×n (G · F).
Tensor Product Model Transformation in Polytopic Model-Based Control by Péter Baranyi