Affine Tensor Product Model Transformation

. 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.


Introduction
The importance of polytopic system descriptions is beyond doubt since the development of influential polytopic modelbased analysis and synthesis methods initially introduced by Boyd et al. in [1].These approaches offer a simple way for stability verification and robust or gain-scheduling controller design via Linear Matrix Inequality (LMI) based methods for polytopic Linear Parameter Varying (LPV) and quasi-LPV (qLPV) models.
TP Model Transformation was introduced as a numerical approach to constructing polytopic TP forms of LPV/qLPV models [2] serving as an alternative to analytical procedures such as the sector nonlinearity technique [3].Furthermore, the separated parameter dependencies within the TP structure can be exploited during the controller design extending the polytopic model-based control analysis and synthesis methods [4,5].
In the past decade, TP Model Transformation has been matured and became an extensive framework within polytopic model-based control [2,3,6].Former related works (e.g., [7][8][9][10][11]) obtained the polytopic TP Model through the HOSVD-based intermediate TP form [12], although the resulting polytopic model does not really benefit from the properties of the HOSVD-based form such as complexity reduction capability and uniqueness.
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].
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.We refer to the new intermediate form as Affine TP Model.

Complexity
The next section discusses the abbreviations and notations used in the paper.Section 3 recalls some concepts of tensor algebra related to polytopic TP modeling; then Section 4 discusses the polytopic form of univariate functions showing its relevance to affine geometrics and introduces the affine SVD.In Section 5, affine SVD is applied to obtain the Affine TP Model.Section 6 describes the application to generate and manipulate polytopic TP Models; then Section 7 shows a simple numerical example.Finally, Section 8 concludes the paper.

Notations
The following abbreviations and notations are used within this paper: , : lower and upper bounds for the  scalar Co(⋅ ⋅ ⋅ ): convex hull (set of all convex comb.).

Basic Concepts
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.
3.1.Tensor Algebra.First, the key definitions and properties of tensor algebra of De Lathauwer et al. [18] are recalled and extended to Hilbert spaces by considering multidimensional arrays on a Hilbert space denoted by  in general.
They can be multiplied with real matrices along the th index that is called -mode tensor product.
Lemma (commutativity of  ̸ = -mode tensor products).Given the tensor A ∈   1 ×⋅⋅⋅×  and the matrices Lemma (multiple -mode tensor products).Given the tensor A ∈   1 ×⋅⋅⋅×  and the matrices U ∈ R ×  , V ∈ R × , one has The inner product and norm are defined.
Definition 4 (inner product and norm of tensors).The inner product ⟨A, B⟩ of tensors A, B ∈   1 ×⋅⋅⋅×  is defined as Then the Frobenius norm of a tensor A is defined as ‖A‖ = √⟨A, A⟩.
To perform other matrix operations (e.g., SVD) along the th index, the tensor can be unfolded to a matrix and restored back to tensor.Definition 5 (-mode unfold tensor).Assume an th-order tensor A ∈   1 ×⋅⋅⋅×  , where the elements can be described on an orthonormal basis with finite  elements; then its mode matrix unfolding is denoted by A () with a size of   × ( +1 ⋅ ⋅ ⋅     1 ⋅ ⋅ ⋅  −1 ) and it contains the th coordinate of a  1 ,...,  element at the position (  ,   ), where then their norms are as follows: ‖c‖ = √⟨c, c⟩.
Along the paper, we will assume that for the considered functions this norm exists and it is finite without mentioning it.
The decomposition will be called (i) orthonormal, if the weighting functions are orthonormal as ⟨  ,   ⟩ =   ∀,  = 1 ⋅ ⋅ ⋅ , (ii) homogeneous, if   (x) = 1 ∀x ∈ X, (iii) polytopic, if the   functions denote convex combinations as Then, in geometric sense, the {c 1 , . . ., c  } vertices construct an enclosing polytope for the image of c(x).Its elements are inside the polytope because they can be described as a convex combination of the vertices.In these cases, letter  will denote the weighting functions through the paper.
where (i) x() denotes the state variables, u() the control inputs, w() the disturbances, y() the measured outputs, and z() the performance outputs, (ii) it is defined on a hyperrectangular parameter domain: (iii) for the sake of brevity, the parameter-dependent system matrices will be denoted as so we have the Ω → S function, where S denotes the space of real matrices with appropriate size.
That is often extended with delayed inputs, delayed states, and so on according to the dynamics of the investigated system; see [19].
Polytopic models are polytopic decomposition of the S(p) system matrix.They are described as convex combinations of so-called vertex system matrices, as and this form allows for using LMI-based control analysis and synthesis methods.

TP Model Transformation. TP Model
Transformation is aimed at transforming the parameter-dependent system matrix S(p) into polytopic form with decoupled parameter dependencies, resulting in a nested parameter-wise polytopic representation that is expressed as multiple tensor products.
Definition 7 (polytopic TP Model).Polytopic TP Models are (q)LPV models with system matrices: 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 (1)   1 ( 1 ) for all  = 1 ⋅ ⋅ ⋅ .
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 [5].

Affine Decomposition of Univariate Functions
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.
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 .Then the elements of the C image can be given as the sum of a value on the (a 1 , . . ., a  ) basis and an (a +1 ) offset, by applying homogeneous coordinates k() as where . With this description, the objects are characterized by coordinates u() = [ 1 (), . . .,   ()] on the affine hull.
Obtaining an enclosing polytope for the u() coordinates in the -dimensional Euclidean space with vertices {r 1 , . . ., r  } as the k() homogeneous coordinates can be expressed as convex combinations of the vertices with weights ( 1 (), . . .,   ()) as and it provides an enclosing polytope for the C image set with the following vertices: because it can be described as their convex combinations: This way, the polytopic description can be constructed for the original image in the  space by considering the dimensional geometric problem.

Affine Singular Value Decomposition of Univariate Functions.
Consider the description on the affine hull in (17) and restrict it to orthogonal (a 1 , . . ., a  ) bases and homogeneous, orthonormal (V 1 (), . . ., V +1 ()) coordinate functions.Then we can define the following unique form that is called Affine Singular Value Decomposition.
Definition 8 (affine SVD (ASVD)).The form represented by ( 17) is called affine SVD of c function if it is a homogeneous, orthonormal decomposition and the a  ∈  ( = 1 ⋅ ⋅ ⋅ ) elements of the basis are orthogonal and ordered by their norms as which are called singular values.The decomposition's uniqueness property is inherited from the uniqueness of SVD.

Lemma (uniqueness of ASVD).
The  1 , . . .,   singular values and the a +1 offset are unique.Now consider the ordered singular values and let ( 1 ,  2 , . ..) denote their multiplicities such that Then the forms and only these forms are valid decomposition where and Q  are arbitrary real orthogonal matrices with size   ×  , respectively.
Proof.These kinds of decomposition are ASVD because (i) by multiplying the orthonormal V  () = 1 ⋅ ⋅ ⋅  functions with a T orthogonal matrix, they remain orthonormal, (ii) by multiplying the orthogonal a  values of the same norm with a Q  orthogonal matrix, they maintain their orthogonality and norm as well.This way, the singular values and their order do not change.
Only this kind of decomposition is ASVD, because (i) to ensure the V  +1 () = 1 and the orthonormality of V  () functions, the offset part cannot change: (ii) the remaining part must be the SVD of function (c() − a +1 ) inheriting its uniqueness properties, which results in the structure of T.
Obviously, if every singular value is different, only the signs of a  objects and V  () functions ( = 1, . . ., ) can be varied, because the lemma allows for only Q  = ±1 values in these cases.

Lemma
(complexity trade-off).Consider the affine SVD in (17) with  singular values, where  is the dimension of the affine hull.
The best  < -dimensional approximation (in terms of the defined norm) can be obtained as Proof.It was shown in (26) that the average value of c function is a +1 so it is the best  = 0-dimensional approximation.
And if the best -dimensional approximation is known, the best  + 1-dimensional can be obtained by adding the a product with maximal possible norm (as in the Eckhart-Young theorem [20]), which is here a +1 V +1 ().
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.
Then s() = k()K  ASVD can be obtained as where the matrices U, S, and V come from the SVD computation:

Definition of Affine Tensor Product Form
This section presents the derivation of polytopic TP forms for multivariate functions c : Ω →  (31) based on the Affine TP form, which represents the affine geometric structure for all parameter dependency, respectively.

Complexity
Remark 13 (ASVD on functions).The definition exploits the fact that functions with norm in Definition 6 constitute Hilbert spaces.This way, the c(p) function can be considered as a univariate function for all  = 1 ⋅ ⋅ ⋅ , where H is the Hilbert space of functions and the ASVD is defined for it.
The polytopic TP form can be obtained by determining enclosing polytopes for all k () (  ) trajectories in the  dimensional spaces for all  = 1 ⋅ ⋅ ⋅  and applying the following theorem.
Proof.From Section 4.1, the uniqueness of the Affine TP form can be characterized by the following theorem.
Proof.Only these forms are allowed by uniqueness properties of ASVD (see Lemma 9) and their -mode expansions show that these forms are ASVD, so the TP form is affine.
The form enables the -mode dimension reductions with the following error (regarding the defined norm) based on the properties of TP forms on orthonormal weighting functions, which are discussed in the Appendix.

eorem (complexity reduction)
. The reduction of one mode dimension from   to    <   with minimal error in the defined norm can be achieved by omitting the (   + 1), . . .,   th subtensors of C  and the corresponding elements of k () (  ).Then the error is The approximation error of dimension reduction in multiple ( = 1 ⋅ ⋅ ⋅ ) parameter dependencies is bounded as Proof.Construct a tensor Ĉaff with the same sizes as C aff that contains zeros in the omitted subtensors.Then, if ΔC aff = C aff − Ĉaff , the approximation error can be written as If only one -mode dimension is decreased, the error of the approximation can be written as (based on Lemma A.2) that is minimal as Lemma 10 indicated.
Considering the case when more than one n-mode dimension is decreased, the worst case (equality) of (40) occurs if there are zero elements in the intersection of the omitted subtensors.Otherwise, the error of the approximation is smaller.
Finally, the method is presented for its exact derivation or at least approximate reconstruction.

Method 17 (numerical reconstruction of Affine TP form).
The first step is to obtain an initial TP form with the desired parameter groups (  ) . (43) Here we describe two approaches for it.
Step 1a (analytical initial form).If the function is analytically given, the c(p) = ĉ(p) initial form may be constructed analytically.
Then the resulting TP form is affine.
Proof.For TP forms on orthonormal weighting functions, if is ASVD as well; see Lemma A.3 of the Appendix.
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 Ω with different interpolation strategies.
Remark 18.The sequential truncation approach (see [26]) can also be applied by using the complexity reductions in iterations of Step 3 in order to decrease the computational cost.

Remark 19. 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.

Application for LPV/qLPV Models
The results of the previous section are appropriate for system matrices S(p) of (q)LPV models (9).By defining the inner product and norm for F, G ∈ S system matrices as the space S constitutes a Hilbert space and the following TP Model can be defined.
Definition 20 (Affine TP Model).The system matrix of the (q)LPV model ( 9) is given in Affine TP form as The elements of core tensor S aff are system matrices and the functions k () (  ) are   -dimensional trajectories given by homogeneous coordinates.
The uniqueness of the description is inherited from Theorem 15.Complexity (dimension of the affine hull) reduction can be done based on Theorem 16 but it must be mentioned that it does not give guarantee about its distribution along the parameter domain in terms of dynamical effects, and thus, it is not closely related to its dynamical properties in ill-conditioned cases.It means that if the omitted details are not only numerical error (representing essential information about the system dynamics), it is recommended to apply robust design methods taking into account the neglected part as in [27].
Furthermore, it has direct link with polytopic model generation based on Theorem 14.
Then the polytopic TP Model ( 13) can be formalized with w () (  ) weighting functions and core tensor There exist numerical methods for enclosing simplex polytope generation (where   =   + 1) such as the Minimal 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).
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.
Then the manipulated polytopic TP Model can be formalized as where Relevant examples are the manipulation of the constraints in MVS method based on the achievable performance with the previous polytopes (see [13,30]) or the nonsimplex method where problematic regions are cut off from the polytope [16].

Numerical Example
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 1.The goal of the control effort is to stabilize its translational motion using a rotational actuator [31][32][33][34][35].
The equation of motion is usually reformulated in the following dimensionless form: where  denotes the dimensionless translational position,  the dimensionless input,  dimensionless time, and  the coupling parameter.The nonlinear ODE is used for the purpose of constructing the qLPV model; the state variables are chosen as and the parameters as Then the constructed qLPV realization reads where  In order to obtain the Affine TP form of function (59), the parameter dependencies are separated leading to an initial TP form with the following weighting functions: performing Step 1a of Method 17.After orthogonalization and sequential ASVD, we get the affine form where the -mode dimensions are  1 = 3 and  2 = 1 and the corresponding weighting functions are depicted in Figure 2. The singular values:  The enclosing polytope generation for  2 dependency is trivial, because it is a one-dimensional problem.The vertices: r (2)  1 = [−1.7311] and r (2)  2 = [1.7311].The three-dimensional problem of  1 dependency is more challenging.The methods for generation of the enclosing polytope can be applied as MVS (or SNNN, CNO); see

Complexity
The resulting polytopic TP Model has two parameter dependencies, and it has the following general form: where 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  = F ( 1 ) x, F ( 1 ) = For more complex examples that apply other polytopic model generation, manipulation methods, and controller design techniques, see papers [9,11,13,16,30].

Conclusion
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.

Complexity 4 . 1 .
Enclosing Polytope on the Affine Hull.Consider the univariate c : [, ](⊂ R) →  function, where  is a Hilbert space.Denote its image to be enclosed by the polytopic form as

)
omitting the zero singular values and the corresponding columns of singular matrices.Proof.v() is orthonormal, because f() is orthonormal and blockdiag (U, 1) is orthogonal.It is homogeneous because V +1 () =   () = 1.The k   values ( = 1 ⋅ ⋅ ⋅ ) are orthogonal and ordered by norm from properties of SVD.

Figure 1 :
Figure 1: The mechanical model of the TORA system.