^{1}

^{1}

This paper introduces the novel concept of Affine Tensor Product (TP) Model and the corresponding model transformation algorithm. Affine TP Model is a unique representation of Linear Parameter Varying systems with advantageous properties that makes it very effective in convex optimization-based controller synthesis. The proposed model form describes the affine geometric structure of the parameter dependencies by a nearly minimum model size and enables a systematic way of geometric complexity reduction. The proposed method is capable of exact analytical model reconstruction and also supports the sampling-based numerical approach with arbitrary discretization grid and interpolation methods. The representation conforms with the latest polytopic model generation and manipulation algorithms. Along these advances, the paper reorganizes and extends the mathematical theory of TP Model Transformation. The practical merit of the proposed concept is demonstrated through a numerical example.

The importance of polytopic system descriptions is beyond doubt since the development of influential polytopic model-based analysis and synthesis methods initially introduced by Boyd et al. in [

TP Model Transformation was introduced as a numerical approach to constructing polytopic TP forms of LPV/qLPV models [

In the past decade, TP Model Transformation has been matured and became an extensive framework within polytopic model-based control [

A recent paper of the authors [

The paper proposes the Affine TP Model that substantially improves the polytopic TP Model generation and manipulation methodology by combining the affine geometric interpretations [

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 [

The next section discusses the abbreviations and notations used in the paper. Section

The following abbreviations and notations are used within this paper:

(q)LPV: (quasi)Linear Parameter Varying

LMI: Linear Matrix Inequality

SVD: Singular Value Decomposition

HOSVD: higher-order singular value decomposition

TP Model: Tensor Product Model

The section briefly discusses the related concepts of tensor algebra, polytopic LPV/qLPV modeling, and the goals of TP Model Transformation introducing the notations that are used in the followings.

First, the key definitions and properties of tensor algebra of De Lathauwer et al. [

They can be multiplied with real matrices along the

The

The definition implies the following properties.

Given the tensor

Given the tensor

The inner product and norm are defined.

The inner product

To perform other matrix operations (e.g., SVD) along the

Assume an

Consider the

The inner product of

Along the paper, we will assume that for the considered functions this norm exists and it is finite without mentioning it.

The decomposition

orthonormal, if the weighting functions are orthonormal as

homogeneous, if

polytopic, if the

Then, in geometric sense, the

Consider the following form of LPV/qLPV models:

it is defined on a hyperrectangular parameter domain:

for the sake of brevity, the parameter-dependent system matrices will be denoted as

That is often extended with delayed inputs, delayed states, and so on according to the dynamics of the investigated system; see [

Polytopic models are polytopic decomposition of the

TP Model Transformation is aimed at transforming the parameter-dependent system matrix

Polytopic TP Models are (q)LPV models with system matrices:

the

the

Let us recall its expanded form and highlight that it is polytopic for all parameter dependencies because the short TP notation can be extended as

It is easy to see that this form is a special polytopic model. This way, the polytopic model-based control analysis and synthesis methodology can apply to them. Furthermore, the parameter separated structure can be exploited during control analysis and synthesis; for more details, see [

The section shows the role of affine geometry in the derivation of polytopic decomposition of univariate functions, and it introduces the Affine Singular Value Decomposition to represent the geometric structure in a unique way that will be applied in the Affine TP Model.

Consider the univariate

Although the considered Hilbert space can be higher dimensional, there may exist polytopic descriptions with a finite number of vertices. It depends on the dimension of the so-called affine hull that is the minimum dimensional affine subspace which contains every object. It can be expressed as the set of affine combinations of the values of the function

The dimension of the affine hull is called affine dimension and denoted by

Obtaining an enclosing polytope for the

This way, the polytopic description can be constructed for the original image in the

Consider the description on the affine hull in (

The form represented by (

The decomposition’s uniqueness property is inherited from the uniqueness of SVD.

The

Now consider the ordered singular values and let

These kinds of decomposition are ASVD because

by multiplying the orthonormal

by multiplying the orthogonal

Only this kind of decomposition is ASVD, because

to ensure the

the remaining part must be the SVD of function

Obviously, if every singular value is different, only the signs of

Consider the affine SVD in (

The best

It was shown in (

And if the best

Because the complexity of enclosing polytope generation depends on the dimension of the affine hull, this property allows for its reduction with minimal error in the defined norm.

The following lemma describes the numerical reconstruction assuming a vector function given as a homogeneous, orthonormal decomposition.

Consider the

Then

This section presents the derivation of polytopic TP forms for multivariate functions

The following form of function (

The definition exploits the fact that functions with norm in Definition

The polytopic TP form can be obtained by determining enclosing polytopes for all

If for all

From Section

The

If (

Only these forms are allowed by uniqueness properties of ASVD (see Lemma

The form enables the

The reduction of one

Construct a tensor

Considering the case when more than one n-mode dimension is decreased, the worst case (equality) of (

Finally, the method is presented for its exact derivation or at least approximate reconstruction.

The first step is to obtain an initial TP form with the desired parameter groups

Then the initial TP form (

Then for index

Then the resulting TP form is affine.

For TP forms on orthonormal weighting functions, if

The method proves the existence of Affine TP forms for cases where the separation of parameter dependencies is possible, and it extends the previous approach by allowing exact analytical separation or the application of discretization with varying density along the parameter domain

The sequential truncation approach (see [

By applying SVD instead of ASVD in Step 3 (and optionally simple orthonormalization in Step 2), the method can be used to determine the so-called HOSVD-based TP form as well.

The results of the previous section are appropriate for system matrices

The system matrix of the (q)LPV model (

The elements of core tensor

The uniqueness of the description is inherited from Theorem

Furthermore, it has direct link with polytopic model generation based on Theorem

The determination of vertices

Then the polytopic TP Model (

There exist numerical methods for enclosing simplex polytope generation (where

Fine-tuning manipulation/optimization is an important technique in polytopic model-based design. Similarly to the polytope generation methods, manipulation techniques are also immediately connectible to the Affine TP Model.

As manipulation of

Then the manipulated polytopic TP Model can be formalized as

Relevant examples are the manipulation of the constraints in MVS method based on the achievable performance with the previous polytopes (see [

This section discusses a control-related example that gives hands-on insight into a realistic design scenario.

Consider the translational oscillator with an eccentric rotational mass actuator (TORA) system shown in Figure

The mechanical model of the TORA system.

The equation of motion is usually reformulated in the following dimensionless form:

The nonlinear ODE is used for the purpose of constructing the qLPV model; the state variables are chosen as

Then the constructed qLPV realization reads

In order to obtain the Affine TP form of function (

The

The enclosing polytope generation for

The three-dimensional problem of

The MVS enclosing polytope for

The

The resulting polytopic TP Model has two parameter dependencies, and it has the following general form:

For the sake of brevity, only the quadratic stabilization via state feedback problem is recalled and applied. To exploit the separated parameter dependencies, the controller-candidate depends only on the first parameter as

Here the problem is feasible with gains:

For more complex examples that apply other polytopic model generation, manipulation methods, and controller design techniques, see papers [

The proposed Affine TP Model Transformation is a significant development in polytopic model-based control providing a general yet practically advantageous methodology for polytopic model generation. The unique Affine TP Model as a central concept serves as starting point for complexity reduction, polytopic model creation, and various polytope manipulation/optimization approaches helping to fully exploit the directly applicable powerful LMI-based synthesis methods. The most important benefits of the proposed intermediate TP form are the geometrically appropriate representation of the LPV structure in each dimension and the capability of dimension reduction with minimal error and low computational cost. In addition to the theoretical discussion, for the sake of technical completeness, an illustrative numerical example was provided to clearly show the practical merit of the Affine TP form.

In some sense, the paper sums up and consolidates the theoretical basis of TP Model Transformation that has been evolved in the past decade through the contribution of a broader research community.

First, the following lemma highlights important properties of orthonormal decomposition.

If

This property appears in orthonormal TP forms in the following way.

If there are two TP functions given on the same orthonormal weighting function system as

This way, the functions’ orthogonality depends only on the orthogonality of the core tensors. Based on this property, the following lemma formalizes an important property of the Affine TP form.

If the weighting functions of TP form

The following form is an ASVD along

The form is an ASVD along

The requirements for the

Conclusively, the

The authors declare that they have no conflicts of interest.

The authors thankfully acknowledge the financial support of this work by the ÚNKP-16-3 and ÚNKP-16-4 New National Excellence Program of the Ministry of Human Capacities and the support of the Doctoral School of Applied Informatics and Applied Mathematics of Óbuda University and Research and Innovation Center of Óbuda University.

_{∞}control for tensor product type polytopic plants