On the approximation properties of TP model forms

Transcription

1 On the approximation properties of TP model forms Domonkos Tikk 1, Péter Baranyi 1 and Ron J. Patton 2 1 Department of Telecommunications and Media Informatics, Budapest University of Technology and Economics H-1117 Budapest, Magyar Tudósok Körútja 2., Hungary s: tikk@tmit.bme.hu, baranyi@tmit.bme.hu 2 Control and Intelligent Systems Research Group, University of Hull Cottingham Road, Hull, HU6 7RX, UK r.j.patton@eng.hull.ac.uk

2 Overview Introduction TP model forms Nowhere denseness theorems and their consequences Trade-off results Conclusion

3 1 Introduction 1.1 TP model forms TP model in broad sense: an approximation technique where the approximating functions are in tensor product form, whereas a TP model form is a particular approximating function in a TP model. TP model in narrower sense: when a TP model is applied to dynamic system control (in focus of this paper). Recently, a large variety of LMI based controller designs have been carried out for TP models under PDC. (TP model transformation). A crucial point of control design frameworks is the modelling accuracy =. This talk analyzes the approximation capabilities of the TP model forms when applied to dynamic system control.

4 1.2 Historical background 1900, Hilbert s 13 th conjecture: there exist such continuous multi-variable functions, which cannot be decomposed as the finite superposition of continuous functions of less variables 1957, Disproved by Arnold and Kolmogorov: Theorem 1 For all n 2, and for any continuous real function f of n variables on the domain [0, 1], f : [0, 1] n R, there exist n(2n + 1) continuous, monotone increasing univariate functions on [0, 1], by which f can be reconstructed according to the following equation 2n n f(x 1,..., x n ) = ψ pq (x p ). (1) q=0 φ q p=1 Problem: functions φ q and ψ pq are often very complicated and highly nonsmooth, so their construction is difficult.

5 Improvements in the 60s by e.g. Sprecher and Lorentz: decreased the number of components. 1980, De Figueiredo: Kolmogorov s theorem can be generalized for multilayer feedforward neural networks these can be considered as universal approximators from late 80s, most neural and fuzzy systems are universal approximators.

6 1.3 Practical limitations Approximating models have exponential complexity in terms of the number of components, i.e. the number of components grows exponentially as the approximation error tends to zero. This exponentiality cannot be eliminated, so the universal approximation property of these models cannot be exploited straightforwardly for practical purposes. 1999, Moser, Tikk: If the number of the components is bounded, the resulting set of models is nowhere dense in the space of approximated functions, i.e., this is an almost discrete sets. (model type: Sugeno controllers).

7 1.4 Possible solution Main question: what extent the approximation should be accurate? Practical aspect: an acceptably good approximation is enough, where the given problem determines the factor of acceptability in terms of the accuracy ε. The task is to find a possible trade-off between the specified accuracy and the number of components. This talk focuses on TP model form based control and its approximation properties due to its importance in system design framework.

8 2 TP model forms 2.1 Dynamic model Assume that the following parametrically varying state-space dynamic model of the system is given: sx(t) = A(p(t))x(t) + B(p(t))u(t) (2) y(t) = C(p(t))x(t) + D(p(t))u(t), The system matrix ( ) A(p(t)) B(p(t)) S(p(t)) = R O I (3) C(p(t)) D(p(t)) is a parametrically varying object, where p(t) Ω is time varying N-dimensional parameter vector, where Ω R N is a closed hypercube. p(t) can also include some elements of x(t). Further, for a continuous-time system sx(t) = ẋ(t); or for a discrete-time system sx(t) = x(t + 1) holds.

9 2.2 Tensor product (TP) model form In this case S(p(t)) is given as a convex combination of the linear vertex systems S 1,...,S I on a compact domain: S(p(t)) co{s 1,...,S I ; w i (p(t))} (4) where w i (p(t)) are the basis functions that defines the convex combination. If the basis is normalized we obtain equivalent form as in TS model case: I I S(p(t)) w i (p(t))s i and w i (p(t)) = 1 i=1 i=1

10 2.3 TP model decomposition form for dimensions S(p(t)) I 1 i 1 =1 I N i N =1 basis functions: w n,in (p(t)) vertex systems: S i1,...,i N Example: ( N ) w n,in (p n (t))s i1,...,i N n=1 ẋ(t) = A(x(t))x(t) + B(x(t))u(t); x(t) = TP model approximation: ẋ(t) I 1 I 2 i 1 =1 i 2 =1 ( ) x1 (t) x 2 (t) w 1,i1 (x 1 (t))w 2,i2 (x 2 (t)) (A i1,i 2 x(t) + B i1,i 2 u(t))

11 3 Nowhere denseness Denotation: matrix function of the following form as TP form of order (I 1, I 2 ) ( ) z1 I 1 I 2 ) TP(z 1, z 2 ) = S w z 1,i1 (z 1 )w 2,i2 (z 2 )S i1,i 2 ( z1 2 z 2 i 1 =1 i 2 =1 where here the number of the component is bounded by I 1 in the first and by I 2 in the second variable. Matrices S i1,i 2 contain the coefficients as defined in (3). w 1,i1 (z 1 ) and w 2,i2 (z 2 ) are the normalized basis functions. TP (p) ( ) (n 1,n 2 ) [0, 1] 2 which is equally a subset of TP forms and L p ( [0, 1] 2). An elements of set (5) is a TP model forms of order (I 1, I 2 ). (5)

12 3.1 Results Lemma 1 Let ω : [0, 1] R be a function of the form ω(z 1, z 2 ) = α if z 2 z 1, ω(z 1, z 2 ) α else, where α R. Then for each p [1, ] and I 1, I 2 N there holds { } inf ω t p t = TP[i,j] TP TP (p) (I 1,I 2 ) > 0. (6) for arbitrary (i, j) pair. Lemma 1 exemplifies a function that cannot be approximated arbitrarily well by TP model forms of order (I 1, I 2 ) w.r.t. ( the L p norm. The ω that is approximated above, thus not en element of cl TP (p) ( )) (I 1,I 2 [0, 1] 2, that is the closure (w.r.t the L p norm) of the subset TP (p) ( ) ( (I 1,I 2 ) [0, 1] 2 of L ) p [0, 1] 2

13 Theorem 1 To each p [1, ], ε > 0 and t = TP [i,j], TP TP (p) ( ) (I 1,I 2 ) [0, 1] 2, I 1, I 2 N there is a continuous function ( ω L p ([0, 1] 2 )\cl TP (p) ( )) (I 1,I 2 ) [0, 1] 2 fulfilling ω t p < ε. Theorem 1 guarantees that to each ε > 0 and t = TP [i,j], TP T (p) ( ) (I 1,I 2 ) [0, 1] 2 ( there is a function ω L p ([0, 1] 2 )\cl TP (p) ( )) (I 1,I 2 ) [0, 1] 2 with ω t p < ε. That is TP (p) (I 1,I 2 ) is no-where dense in Lp ([0, 1] 2 ).

14 4 Examples u Two examples are presented by means of a mechanical system of Figure The original non-linear state-space model of the system that can exactly be represented by a TP model; 2. Same system with modified nonlinear term. Exact model does not exist. m x k d Figure 1: Mass spring damper system

15 4.1 Original model The dynamical equation of the mechanical system of Figure 1 is given as: m ẍ(t) + g(x(t), ẋ(t)) + k(x(t)) = φ(ẋ(t)) u(t), (7) where m is the mass and u(t) represents the force. The function k(x) is the nonlinear or uncertain stiffness coefficient of the spring, g(x, ẋ) is the non-linear or uncertain term damping coefficient of the damper, and φ(ẋ(t)) is the nonlinear input term. Assume that g(x(t), ẋ(t)) = d(c 1 x(t) + c 2 ẋ 3 (t)), k(x(t)) = c 3 x(t) + c 4 x 3 (t), and φ(ẋ(t)) = 1 + c 5 ẋ 3 (t). TP model transformation is performed. If the density of sampling grid is increased to infinity the limes of the rank of the sampled tensor remains 2 on the first two dimensions. Exact representation is possible.

16 Basis functions Basis functions x x 2 Figure 2: The basis functions obtained by TP model transformation; 2 functions in each dimension assure exact approximation

17 4.2 Modified system where g(x, ẋ) = d(c 1 x f(x, ẋ)ẋ +c 2 ẋ 3 ) f(x 1, x 2 ) = { 0.5 if x 2 x (x 1 x 2 ) if x 2 < x 1 is a continuous function constructed based upon the ω in Lemma x Figure 3: The graph of function f(x 1, x 2 ) over the region [ 1.5, 1.5] [ 1.5, 1.5] x

18 Approximation depends on the density of the sampling grid. In the previous example number of basis functions does not increase with the increase of sampling points. In the case of the current example the necessary number of basis functions tends to infinity with the increase of the number of measurement points Table 1 shows the maximum error of the approximation when the function was sampled on a equidistant grid on the parameter space. Table 1: Maximum error vs. number of basis functions in each dimension number of basis functions maximum error

19 Basis functions Basis functions x x 2 Figure 4: 50 basis functions in each dimension the model is approximated by TP model form

20 5 Conclusion Theorem 1 points out if we restrict the number of components the corresponding set of TP model forms is nowhere dense in the approximation model space. The solution for this discrepancy can be the trade-off between the accuracy and the number of components.