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Design Methodology of Tensor Product Based Control Models via HOSVD and LMIs Domonkos Tikk∗ , P´eter Baranyi∗ , Ron J. Patton∗∗ , Imre Rudas∗∗∗ and J´ozsef K. Tar∗∗∗ Dept. Telecommunication and Telematics, Budapest University of Technology and Economics H-1117 Budapest, Magyar Tud´osok k¨or´utja 2., Hungary E-mail: tikk/[email protected] ∗∗ Control and Intelligent Systems Research Group, University of Hull, Cottingham Road, Hull, HU6 7RX, UK E-mail: [email protected] ∗∗∗ Department of Information Technology, B´anki Don´at Polytechnic, H-1081 Budapest, N´epsz´ınh´az utca 8., Hungary E-mail: rudas/[email protected] ∗

Abstract This paper aims at solving the conflicts of the computational needs of building Tensor Product (TP) based control models having high approximation accuracy and practical aspects of their applicability w.r.t. system stability and feasibility. Therefore first we propose a HOSVD (Higher Order Singular Value Decomposition) based methodology capable of implementing any differential equation systems of a dynamic model in TP model form with specific basis functions whereupon the Linear Matrix Inequalities (LMI) based controller design techniques and stability analysis can be executed. Second, we intend to find a tradeoff between the TP modelling accuracy, hence system performance, and the controller complexity which is bounded by the available real time computation power at hand. As an example a detailed control design is given. Keywords: TP based approximation, HOSVD based reduction, LMI control design, accuracy–complexity tradeoff. 1. Introduction The recent results in control theory provides very fast solutions for many convex optimization problems for which no traditional “analytic” or “closed form” solution are known. This fact has far-reaching implications for engineers: it changes our fundamental notion of what’s being a solution to a problem. There are “analytical solutions” to a few very special cases in the wide variety of problems in systems and control theory, but in general non-linear problems cannot be solved. A number of problems that arise in systems and control, e.g. optimal matrix scaling, digital filter realization, interpolation problems that arise in system identification, robustness analysis and state feedback synthesis via Lyapunov functions, can be reduced to a handful of standard convex and quasiconvex problems that involve Linear Matrix Inequalities (LMI). Extremely efficient interior point algorithms have recently been developed for and tested on these standard problems. With regard to the above the use of Takagi–Sugeno (TS) modelling [1] is motivated by the fact that it can easily be cast or recast as convex optimization which involves LMIs [2, 3, 4, 5]. Therefore controller design and Lyapunov stability analysis of a non-linear system can be reduced to solving an LMI problem in many cases.

Based on this idea Tanaka et al. [2, 3, 4] and Wang et al. [5] have introduced various powerful controller design methods and relaxed stability theorems. A quasi standard controller design methodology has been simultaneously proposed to polytopic models. Polytopic and TS models analytically have the same explicit form. Both of them are a kind of convex combination or superposition of linearized models. These models can be commonly treated in the framework of standard Tensor Product (TP) based approximation, where the approximation points become linear models. In order to facilitate further reading let the polytopic and TS models be termed as TP model. Since the TP model has the universal approximation property, we may conclude that any kinds of differential equation systems can be given in TP model form, whereupon the execution of numerical techniques, such as LMI, leads to a controller design and stability analysis. Despite this advantageous idea discussed above the use of TP models is practically limited. The reason of this is that though the TP model has the universal approximation property in theory [6, 7], it suffers from the problem of exponential complexity with regard to approximation accuracy. It means that the number of linearized models in the convex combination grows exponentially as the approximation error tends to zero. Moreover, for TP model it is shown, that if the number of linearized models is bounded, the resulting set of function is nowhere dense in the space of approximated functions [8, 9, 10]. In conclusion, the exponential complexity explosion of TP models cannot be eliminated, so the universal approximation property of these modelling approaches cannot be exploited straightforwardly for practical purposes. The mutually contradicting results naturally pose the question as to what extent the model approximation is deemed accurate enough for the available complexity. From the practical point of view, though, it is enough to achieve an “acceptably” good approximation, where the given problem determines the factor of acceptability in terms of the acceptable error. The task is, hence, to find a possible trade-off between the specified accuracy and the complexity. Motivated by the above conflicts the main objective of this paper is twofold. The first is to propose a HOSVD based methodology capable of implementing any differential equation systems of a dynamic model in TP model

form with specific basis functions whereupon the LMI based controller design techniques and stability analysis can be executed. The second is to find a tradeoff between the TP modelling accuracy, hence system performance, and the controller complexity which is bounded by the available real time computation power at hand. The previous works [11, 12] of this paper gives detailed introduction to the concepts utilized in this paper especially to the aspects of HOSVD in fuzzy rule base complexity reduction. 2. TS model and restricted approximation property in practice Dynamical system models of plants operating around particular operating conditions are often obtained as multimodel representations of the form λx(t) = Ai x(t)+Bi u(t) y(t) = Ci x(t)+Di u(t); (1) where λx(t) = x(t) ˙ for a continuous-time system or λx(t) = x(t + 1) for a discrete-time system. The multimodel (1) can be alternatively specified by the sequence of system matrices
 Ai B i Si = (2) Ci Di Such a model belongs to the more general class of socalled polytopic models λx(t) = A(p)x(t) + B(p)u(t) y(t) = C(p)x(t) + D(p)u(t); whose time-varying system matrix
 A(p) B(p) S(t) = , C(p) D(p)

(3)

The explicit form of TS model (4) can be reformulated in terms of tensor product as:  N   S(p) = S wn (pn ) , (5) n=1

(for definition and notation of tensor product see Appendix) where the row vector wn (pn ) ∈ RIn contains the basis functions wn,in (pn ), the N + 2 -dimensional coefficient tensor S ∈ RI1 ×I2 ×···×IN ×O×L is constructed from the matrices Si1 ,i2 ,...,iN ∈ RO×L . The first N dimensions of S are assigned to the dimensions of the parameter space P . The next two are assigned to the output and input vectors, respectively. In order to facilitate further reading, let us define conditions SN and NN, which characterize the normalized basis functions wn,i (xn ) as: Property 1: (Non-Negativeness: NN) Basis function w(x), x ∈ X, has the non-negativeness property if ∀x : w(x) ∈ [0, 1]. For further reading let us extend this definition for matrices: a given matrix U is called non-negative if all of its elements are bounded by [0, 1]. Property 2: (Sum-Normalization: SN) Basis functions wi (x),i = 1..I, x ∈ X, have the sum-normalization propI erty if they fulfill: ∀x : i=1 wi (x) = 1. For matrices: a given matrix U is called sum-normalized if (U) = 1 (where  operator (A) sums up the column vectors of A as: (A) = A1). Property 3: (Normalization: NO) Basis function w(x), x ∈ X, has the normalization property if ∃x : w(x) = 1;. For matrices: a given matrix U is called NO if its every column has at least one element with value one. Without the loss of generality let us consider a two dimensional problem as a demonstration. Suppose that the following dynamic model is given:

where p is time varying, may include some elements of x(t). The system matrix varies within a fixed polytope of matrices S(t) ∈ {S1 , S2 , . . . , SN } and S(t) = N N i=1 wi Si : wi ≥ 0, i=1 wi = 1}. The systems S1 , S2 , . . . , SN are called vertex systems. If the convex combination of the vertex system matrices are defined by basis functions w(p) given over p then we obtain the explicit form of TS-fuzzy model:

I  J 

N 

i=1 j=1

S(p) =

wi (p)Si

with i=1 wi (p) = 1, wi (p) ≥ 0}. When the basis functions are decomposed for dimensions we deal with a higher order structure as: S(p) =

I1  I2  i1 =1 i2 =1

..

IN  N 

x ¨(t) = f (x(t), ˙ x(t), u(t)) This equation can be approximated over a compact domain by a convex combination of linearized models: x ¨(t) =

(6)

w1,i (x(t))w ˙ ˙ + bi,j x(t) + ci,j u(t)). 2,j (x(t))(ai,j x(t)

In state space form

i=1

N

(N +1)

x˙ =

I  J 

w1,i (x1 )w2,j (x2 ) (Ai,j x + Bi,j u)

i=1 j=1

which takes the form wn,in (pn )Si1 ,i2 ,..,iN

(4)

iN =1 n=1

where values pn are the elements of p. As a matter of fact, the basis is normalized as above: {∀n : In in =1 wn,in (pn ) = 1, ∀i, n, pn : wn,i (pn ) ≥ 0}. Equation (4) is equivalent with the explicit form of MISO TS (Takagi-Sugeno ) fuzzy model defined by fuzzy rule: IF antecedent set w1,i1 AND antecedent set w2,i2 . . . AND antecedent set wN,iN THEN Si1 ,i2 ,...,iN

x˙ = A ×1 w1 (x1 ) ×2 w2 (x2 ) ×4 xT + + B ×1 w1 (x1 ) ×2 w2 (x2 ) ×4 uT =

x˙ = S ×1 w1 (x1 ) ×2 w2 (x2 ) ×4 xT uT . Zero approximation error is achieved, in general case, if I and J goes to infinity. They, however, becomes constant I, J = C1 , C2 < ∞ if a specified approximation error ε is acceptable. If ε is numerically zero then the right side and the left side of Eq. (8) is considered to be numerically

equivalent, which implies for a large variety of engineering applications that any function can be computed in the form of Eq. (6) by computer in finite time. Here we remark an interesting result from [9]. If the number of basis functions wn,i (xn ) in each input dimension, and hence, the number of models is bounded by I, J, and I · J, respectively, then the set of TS approximators in the form (4) is not capable of approximating with arbitrary accuracy a particular continuous function. As a consequence, the set of TS approximators with bounded number of basis functions is nowhere dense in the space of continuous functions with respect to the Lp norm p ∈ [1, ∞]. 3. From differential equations to complexity minimized TP model This section proposes a HOSVD based method capable of determining the tradeoff between approximation accuracy and complexity. It transforms the differential equations of a dynamic model given over a compact domain via numerical algorithm without analytical derivations, i.e. it automatically gives a TP representation over a minimal complexity according to a given modelling error. Before starting with the method, let us briefly recall some basic concepts utilized in this paper. The proposed method uses an N dimensional hyper rectangular sampling grid. In the followings we require the grid to be uniformly distributed in a bounded space Ω: Definition 1: Let RN ⊃ Ω = [a1 , b1 ] × · · · × [aN , bN ], further let {λn }∞ k=1 be a sequence of finite subsets of Ω with #λk = k. If    #(λk ∩ ω) |ω|   0 ∃k0 ∀ω ∈ Ω ∀k ≥ k0 :  − #λk |Ω|  (7) then the set λk are uniformly distributed on the domain Ω. Here #(λk ∩ ω) denotes the cardinality of the finite set (λk ∩ ω), and |ω| is the Lebesque measure of ω. From now on, the density of the uniformly distributed grid points, defined by an = (an,1 , an,2 , . . . , an,In ), is expressed by value λk (in the sense of Definition 1). Consequently, in our denotation, k, the number of sample points N equals n=1 In . Let us recall the theorem of HOSVD: Theorem 1: (Higher Order SVD (HOSVD)) Every tensor A ∈ RI1 ×I2 ×···×IN can be written as the product A=S

N 

Un

(8)

n=1

in which  1. Un = u1,n u2,n . . . uIN ,n is a unitary (IN × IN )matrix called n-mode singular matrix. 2. tensor S ∈ RI1 ×I2 ×...×IN whose subtensors Sin =α have the properties of (i) all-orthogonality: two subtensors Sin =α and Sin =β are orthogonal for all possible values of n, α and β : Sin =α , Sin =β  = 0 when α = β, (ii) ordering: Sin =1  ≥ Sin =2  ≥ · · · ≥ Sin =In  ≥ 0 for all possible values of n. (n) The Frobenius norm Sin =i , symbolized by σi , are n-

mode singular values of A and the vector ui,n is an ith singular vector. S is termed core tensor. More detailed discussion of HOSVD is given in [13]. A detailed algorithm of HOSVD is given in papers [11, 12]. Papers [11, 12] propose additional transformations to HOSVD pertaining to SN, NN and NO conditions of the basis matrices Un . Let the goal of these transformations be specified by the following theorem: Theorem 2: (SN, NN and NO basis matrices) Assume that executing HOSVD on tensor B ∈ RI1 ,I2 ,...,IN , results N

in B = S ⊗ [ Un Udn ], where the size of matrix Un is n=1

(In × Inr ). Discarding of singular values and their correr sponding singular vectors stored in Udn ∈ RIn ×(In −In ) , N where “d” denotes “discarded”, yields B = B r ⊗ Un . n=1

Matrix Un can always be transformed to SN and NN matrix, but cannot always be transformed to NO. However decreasing the number of discarded singular values, i.e. increasing the number of basis functions makes possible to transform to NO. If the number of basis functions is limited then, at least, we can transform to NO as close as possible. Un subject to N

B r ⊗ Un = B n=1

r N

⊗ Un .

n=1

(9)

Notation Un denotes here that Un is SN, NN and NO r normalized, further, B means that Br is modified subject to equ. (9). Let us turn back to the main objective of this Section. Method 1: Assume that the differential equation of a dynamic model is given in the form of
 ˙ x(t) = f (x(t), u(t), p(t)). (10) y(t) Again, vector p(t) ∈ RN varies in the parameter space P in which the given system is nonlinear. The goal of this method is to extract tensor product based approximation
 ˙ x(t) = f (x(t), u(t), p(t)) ∼ (11) = y(t) r ∼ = (S

N 



wrn (pn (t))) ×N +2 xT (t) uT (t) ,

n=1

where the size of vector wrn (pn (t)) is minimized and it is SN and NN. Step 1: Let (10) be sampled over a high density rectangular grid defined by grid points pi1 ,i2 ,...,iN = ai1 ai,2 . . . aiN : f (x(t), u(t), pi1 ,i2 ,...,iN ) =

= Si1 ,i2 ,...,iN ×2 xT (t) uT (t) ,

(12)

Having the sampled matrices a simple bi-linear approximation can be defined by a first order, i.e. triangular shaped, basis, see figure 1:
 ˙ x(t) = f (x(t), u(t), p(t)) ∼ (13) = y(t)

∼ = (S

N 



wn (pn )) ×N +2 xT (t) uT (t) ,

n=1

Along in the same line as in the previous section, the density λ of the grid points is increased until an acceptable εa is achieved. Step 2. Let HOSVD be executed on tensor S in such a way that the SVD is performed only on dimensions n ∈ [1, N ]. Dimensions N + 1 and N + 2 are not reduced. Therefore one obtains:
 ˙ x(t) = f (x(t), u(t), p(t)) ∼ (14) = y(t)   N 

r r ∼ w (pn ) ×N +2 xT (t) uT (t) , = S n

n=1

where S

N 

 wn (xn ) =

n=1

= Sr

S

r

N 

 Un

n=1 N 

N 

wn (xn ) =

n=1

(wn (xn )Un ) = S r

n=1

N 

λk :k→∞

wnr (xn ),

n=1 r In

where row vector wnr (xn ) ∈ R contains the extracted basis. The number of extracted basis functions on the nth input dimension equals the n-mode rank of tensor S. In this light we can conclude that a minimal basis is obtained. The extracted basis functions are the linear combinations of the original basis as: wnr (xn ) = wn (xn )Un .

(15)

Observe that the extracted basis functions are piecewise linear. The joining points of the linear pieces are located over the grid lines. This comes from the fact that the original basis functions, wn,i (xn ), have the same property. This immediately leads to the following: r wn,j (an,in ) = uin ,j ,

they are not equal to the dynamic system over any operating points if the basis functions do not contain values one. Namely the vertex models has no relation to the real system. In this light they are fictive models, and may be not controlable. For instance the vertex models generated without SN and NN are orthogonal. This means that the LMI design may fail in cases. If the vertex models are identical with the real system over specified operation point, namely, the basis functions reach value 1 at least over one point then we may get closer to the real system in LMI design. Therefore let us extend HOSVD with the NN, SN and NO transformation which yields: 
   N

r ˙ x(t) r ∼ wn (pn ) ×N +2 xT (t) uT (t) . = S y(t) n=1 If the rank of tensor S sampled from the differential equations tends to a constant while we increase λk , the density of the hyper rectangular grid, to infinity, then we can state that the “rank” of the given system is finite: ∀n : rankn (system) = lim rankn (S) = Cn . (17)

(16)

and between the characteristic points (an,in , uin ,j ) the r extracted functions wn,j (xn ) has linear pieces. Consequently, the number of basis functions equals the number of columns in the singular matrix Un , and the values of the basis functions over the grid lines equal the elements of the columns in the singular matrix, respectively. Thus, the more dense grid we define, the more precise basis functions we obtain, and the size of tensor S r increases minimally. The basis functions defined by the column vectors of Un resulted by HOSVD may not be interpretable as SN and NN basis, because the bound of the elements in Un is [−1, 1]. Another crucial point is that the resulted basis functions do not guarantee property SN. This fact would also destroy the effectiveness of the whole concept if the resulted tensor product form is for further studies in, for instance, LMI stability analysis, where the basis functions assumed to have property SN and NN as discussed in the previous Section. The use of NO is not always important. If we analyze the linear vertex systems we find that

This implies that the given function can exactly be transformed to tensor product based form. If the rank of S goes to infinity then the tensor product representation, which has finite number of terms, obviously becomes an approximation. If the number of larger singular values, which are not considered to be numerically zero, tends to constant and only the number of small values may increase with the density of the grid, then we obtain a quantity, which can be considered the numerical rank of the given differential function of the dynamic system. The last important issue of this section that should be discussed is the trade-off between the size of tensor S r and the approximation accuracy. Discarding non-zero singular values in HOSVD is equivalent with decreasing the rank of a given tensor in such a way that its L2 norm changes minimally. Papers [11, 12, 13] show that the maximum error between the elements of S and its lower ranked variN ant S r ⊗ Un is the sum of all the discarded singular valn=1

ues. This implies the error bound of the proposed HOSVD technique: Theorem 3: (Error bound) Assume that an approximation N

of a system is given as S ⊗ wn (xn ), whereupon executn=1

ing the proposed HOSVD technique in non-exact mode, i.e. we discard non-zero singular values as well, yields N

S r ⊗ wnr (xn ). The error bound is characterized as: n=1

N

N

n=1

n=1

εr = max |S ⊗ wn (xn ) − S r ⊗ wnr (xn )| = Φ, (18) x

where, Φ is the sum of the discarded singular values. 4. Example This Section is intended to show the use of the proposed HOSVD based linearization technique via a demonstrative example. Assume that a mechanical system depicted on Figure 2 is given. Its dynamical equation is given as: m·x ¨(t) + g(x(t), x(t)) ˙ + k(x(t)) = φ(x(t)) ˙ · u(t) (19)

w (x)

wn 1,(x)

an 1,

an 2,

wn , (I x) n

an 3,

an , I- 1 n

an , I n

Xn

Fig. 1. Triangular shaped antecedents where m is the mass and u represents the force. Function k(x) is the non-linear or uncertain stiffness coefficient of the spring, g(x, x) ˙ is the non-linear or uncertain term damping coefficient of the damper, and φ(x(t)) ˙ is the non-linear input term. Assume that g(x(t), x(t)) ˙ = d(c1 x(t) + c2 x˙ 3 (t)), k(x(t)) = c3 x(t) + c4 x3 (t), and φ(x(t)) ˙ = 1 + c5 x˙ 3 (t). Furthermore, assume that x ∈ [−a, a], x(t) ˙ ∈ [−b, b] and a, b > 0. The above parameters are set as follows [5]: m = 1, d = 1, c1 = 0.01, c2 = 0.1, c3 = 0.01, c4 = 0.67, c5 = 0, a = 1.5, and b = 1.5. Equation (19) then becomes: x ¨(t) = −0.1x˙ 3 (t) − 0.02x(t) − 0.67x3 (t) + u(t). (20)

Executing Method 1 in exact mode on matrices Ai,j , namely, on tensor A ∈ R400×400×2×2 (note that matrices Bi,j are constant) results in two non-zero singular values to both dimension. Note that if we increase the density of sampling grid to infinity the limes of the rank of the sampled tensor will be 2 on the first two dimensions (actually, the proof of this statement is the analytic derivation of the dynamic model to TP, see [5]. This means that two basis functions on each dimension are sufficient, which is in full accordance with the analytic derivation given in [5], furthermore, the obtained basis is the same (in numerical sense) as the analytically derived one in [5]. Consequently, the TP model is: ∼ A(x(t), x(t)) ˙ =

2  2 

r r r w1,i (x(t))w2,j (x(t))A ˙ i,j .

i=1 j=1

The next step is the a controller design to the extracted TP model. Applying the same LMI technique as utilized in [5], the resulted control signal takes the form: u(t) = −

2  2 

r r w1,i (x(t))w2,j (x(t))F ˙ i,j ,

i=1 j=1

u

where the feedback gains Fi,j are given by LMI approach.

m x d

k

Fig. 2. Mass-spring-damper system The non-linear terms are −0.1x˙ 3 (t) and −0.67x3 (t). In the next part we assume that the analytic derivation from the model (19) to a TP model (5) is unknown like in usual practical cases. According to the proposed method, we sample the differential equations over the dense (400 × 400) approximation grid points defined by xi = −1.5 + (i − 1)3/399 and x˙ j = −1.5 + (j − 1)3/399. Thus the bi-linear approximation with simple first order basis becomes x ¨(t) ∼ (21) = 400  400 

w1,i (x(t))w2,j (x(t))(a ˙ ˙ i,j x(t)+b i,j x(t)+ci,j u(t)),

i=1 j=1

where ai,j = −0.1(−1.5 + (j − 1)3/399)2 , bi,j = −0.02 − 0.67(−1.5 + (i − 1)3/399)2 , and ci,j = 1. In matrix form: 400  400  ∼ A(x(t), x(t)) ˙ w1,i (x(t))w2,j (x(t))A ˙ = i,j ; i=1 j=1

B(x(t), x(t)) ˙ =


 1 . 0

Fig. 3. Control result. x˙ and x are depicted by dash and solid line respectively. The control signal is depicted by dash dot line. Figure 3 shows the effect of the controller, where the stability point is achieved in about 8s. Figure 4 shows a case where white noise is added to signals u, x˙ and x. The white noise has nonzero mean value, which is 10% of the original signal, and its maximum amplitude is 20% of the original signal. 5. Conclusions The main novelty of this paper is to propose a methodology capable of automatically transforming, according to a given approximation accuracy, any differential equation systems of dynamic models given over a compact domain to TP model form, which can easily be formed as convex optimization and whereupon LMI based control design can, hence, be performed. Automatically means here that the TP form is obtained without problem dependent

are given by A ×n U = B, where B(n) = U · A(n) . Let N

A ×1 U1 ×2 U2 × · · · ×N UN be denoted as A ⊗ Un n=1

for brevity. Definition 4: (n-mode rank of tensor A) The n-mode rank of A, denoted by Rn = rankn (A), is the dimension of the vector space spanned by the n-mode matrices as rankn (A) = rank(A(n) ).

white noise Controller

u(t) x(t)

System

y(t)

white noise

Fig. 4. Control result. x˙ and x are depicted by dash and solid line respectively. The control signal is depicted by dash dot line. analytical derivations. This paper also points out an important fact as the motivation of the present approach, that the widely adopted linearization by TP forms can not arbitrarily well describe a given model, it needs to consider the tradeoff between complexity and approximation accuracy. The proposed method finds a minimized TP form, which leads to a complexity reduced controller and controller design. Acknowledgments This research was supported by the Hungarian Scientific Research Fund (OTKA) Grants No. D034614, F30056, T34212 and T34233, and by the Hungarian Ministry of Education Grant No. FKFP 0180/2001. P. Baranyi is supported by the Zolt´an Magyary Scholarship. Appendix Definition 2: (n-mode matrix of tensor A) Assume an ×I2 ×...×IN N th order tensor A ∈ RI1 . The n-mode maIn ×J trix A(n) ∈ R , J = k Ik contains all the vectors in the nth dimension of tensor A, where k = 1, . . . , N and k = n. The ordering of the vectors is arbitrary in A(n) . This ordering shall, however, be consistently used later on. (A(n) )j is called an jth n-mode vector. Note that any matrix of which the columns are given by nmode vectors (A(n) )j can readily be restored to become tensor A. The restoration can be executed even in case when some rows of A(n) are discarded since the value of In has no role in the ordering of (A(n) )j [13, 14]. Definition 3: (n-mode matrix-tensor product) The nmode product of tensor A ∈ RI1 ×I2 ×···×IN and a matrix U ∈ RJ×In , as denoted by A ×n U, is an (I1 × I2 × · · · × In−1 × J × In+1 × · · · × IN )-tensor of which the entries

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