The Lower Bound Error for polynomial NARMAX using an Arbitrary Number of Natural Interval Extensions
Priscila F. S. Guedes, M. L. C. Peixoto, A. M. Barbosa, S. A. M. Martins, E. G. Nepomuceno
DDINCON 2017
CONFERˆENCIA BRASILEIRA DE DIN ˆAMICA, CONTROLE E APLICAC¸ ˜OES30 de outubro a 01 de novembro de 2017 – S˜ao Jos´e do Rio Preto/SP
The Lower Bound Error for polynomial NARMAX using anArbitrary Number of Natural Interval Extensions
Priscila F. S. Guedes M´arcia L. C. Peixoto Al´ıpio M. Barbosa Samir A. M. Martins Erivelton G. Nepomuceno Control and Modelling Group (GCOM), Department of Eletrical Engineering,Federal University of S˜ao Jo˜ao del-Rei, MG, Brazil Centro Universit´ario Newton Paiva, Belo Horizonte, MG, Brazil
Abstract . The polynomial NARMAX (Nonlinear AutoRegressive Moving Average modelwith eXogenous input) is a model that represents the dynamics of physical systems. Thispolynomial contains information from the past of the inputs and outputs of the process,that is, it is a recursive model. In digital computers this generates the propagation of therounding error. Our procedure is based on the estimation of the maximum value of thelower bound error considering an arbitrary number of pseudo-orbits produced from differentnatural interval extensions, and a posterior Lyapunov exponent calculation. We appliedsuccessfully our technique for two identified models of the systems: sine map and Duffing-Ueda oscillator.
Keywords . Polynomial NARMAX, Lower Bound Error, Natural Interval Extension, Inter-val Arithmetic.
System identification is one of the most consolidated and relevant fields of study inscience. One of the aims of this science is to obtain mathematical models analogous tothe phenomena observed in nature. By analogous systems is meant a system capable of pri12 [email protected] [email protected] [email protected] [email protected] [email protected] a r X i v : . [ ee ss . SP ] N ov reproducing the characteristics observed in nature. With the identification of systems it ispossible to model and investigate systems in an attempt to find some pattern in the obser-vations [1,3]. To identify a system, is necessary to assume a model capable of representingthe linear and nonlinear characteristics of the system. Models are mathematical equa-tions that try to describe an approximation of the real system. In the literature there areseveral ways to identify the same system [10]. Different mathematical and computationalrepresentations are used, it can be mentioned the neural networks, fuzzy logic, NARMAX(Nonlinear AutoRegressive Moving Average model with eXogenous input) models, amongothers. The representation of nonlinear systems can be obtained by means of the polyno-mial NARMAX [4]. The nonlinear NARMAX polynomials are linear in the parameters,which allows the use of parameter estimation algorithms for linear models [5]. This math-ematical representation can be seen as a well-organized recursive function in which theparameters are cautiously chosen [5].In general, little attention has been given to the propagation of error in the area ofsystem identification, especially in situations that present the polynomial NARMAX. Oneof the first works related to this subject was [8]. The authors have presented a theorem toestimate the lower bound error for the polynomial NARMAX. In that work, two pseudo-orbits from two different interval extensionswere used to estimate the lower bound error.And in [9], it was found that the basin of attraction and the invariant distribution werenot preserved, the authors shows that from two natural interval extensions may resultin differents trajectories, using the lower bound error. Thus, this work aims to estimatethe lower bound error for n pseudo-orbits derived from n different interval extensions.The proposed method is applied in two identified models of the systems: sine map andDuffing-Ueda oscillator. Afterwards, the Lyapunov exponent, which is a parameter thatcharacterizes the attractor dynamics. It measures the rate of divergence of neighboringorbits within the attractor, quantifying the dependence or sensitivity of the system relativeto the initial conditions [11], is calculated and compares this result with the present in theliterature for the method presented here and for the one proposed by [6].The rest of the paper is organized as follows. In Section 2 we recall some preliminaryconcepts of lower bound error and representation the nonlinear systems. Then, in Section3, we present the developed method. Section 4 is devoted to present the results, then thefinal remarks are given in Section 5. The NARMAX model is a representation for nonlinear systems. This model can berepresented as [4] y ( k ) = F l (cid:2) y k -1 , · · · , y k - n y , u k -1 , · · · , u k - n u , e k -1 , · · · , e k - n e (cid:3) + e k , (1)where y k , u k e e k are, respectively, the output, the input and the noise terms at the discretetime n ∈ N . The parameters n y , n u e n e are their maximum lag. And F (cid:96) is a nonlinearfunction of degree (cid:96) . In recursive functions is possible to calculate the state x n +1 , at a give time, from anearlier state x n x n +1 = f ( x n ) , (2)where f is a recursive function and x n is a function state at the discrete time n. Givenan initial condition x , successive applications of the function f it is possible to know thesequence { x n } . This sequence can be represented by { x n } = [ x , x , · · · , x n ] and is definedas an orbit.Using the computer to calculated the recursive functions, numeric errors are propa-gated during successive calculations, then the true orbit is not calculated but a represen-tation of the same that is called pseudo-orbit. { ˆ x i,n } = [ˆ x i, , ˆ x i, , · · · , ˆ x i,n ] such that | x n − ˆ x i,n | ≤ δ i,n (3) where δ i,n ∈ R is an error and δ i,n ≥
0. So, we may define an interval associated witheach value of a pseudo-orbit I i,n = [ˆ x i,n − δ i,n , ˆ x i,n + δ i,n ] . Thus x n ∈ I i,n for all i ∈ N . (4) The natural interval extension is achieved by changing the sequence of arithmeticoperation [7], that is, the extensions are mathematically equivalents.Furthermore, two extension which algebraically is the same function may not be equiv-alent in interval arithmetic.
The lower bound error was proposed by [8]. It is a practical tool capable of increasingthe reliability of the computational simulation of dynamic systems.
Theorem 1.
Let two pseudo-orbits { ˆ x a,n } and { ˆ x b,n } derived from two interval extensions.Let δ α,n = | ˆ x a,n − ˆ x b,n | / be the lower bound error of a map f ( x ) , then δ a,n ≥ δ α,n or δ b,n ≥ δ α,n . The proof of this theorem can be found in [8].
This section is an extension of the work of [8]. The authors developed the Lowerbound error theorem for two pseudo-orbits from two different interval extensions. But, fora same map may exist more than two natural interval extensions, so the objective is toinvestigate the behavior of the natural interval extensions in the computer, exploring theeffect of interval dependence, due to the repeated presence of a same interval variable inan algebraic expression, then check on the lower bound error of the pseudo-orbits derivedfrom the different natural interval extensions and calculate the Lyapunov exponent of thelower bound error from n pseudo-orbits and compare this value. It was clear that thefunction has more than two extensions, that is, can be rewritten in different ways.The proposed method can be summarized in the following steps:1. Choose the natural interval extensions;2. Calculate the sequence of points of each system from the chosen extensions;3. Determine the lower bound error from the combination of two functions;4. Determine the maximum lower bound error;5. Calculate the Lyapunov exponent and compare the result with the present in liter-ature. The generalization of the lower bound error is presented in the following theorem.
Theorem 2.
Let an arbitrary number k ∈ Z + of pseudo-orbits derived from intervalextensions. ζ n = max | (ˆ x i,n − ˆ x j,n ) | is the lower bound error, subject to i (cid:54) = j , i, j ∈ N , i ≤ k and j ≤ k .Proof. The proof is conducted by reduction ad absurdum . Conversely, let the distancebetween two pseudo-orbit given by γ n = | (ˆ x i,n − ˆ x j,n ) | and let us assume that it is possibleto have a lower bound error described by β n < max | (ˆ x i,n − ˆ x j,n ) | . Then , I i,n = [ˆ x i,n − β i,n , ˆ x i,n + β i,n ] and I j,n = [ˆ x j,n − β j,n , ˆ x j,n + β j,n ] , for all i and j . If it is true, considering the two pseudo-orbits, let us say, a and b , forwhich we have maximum distance between them, it implies that I a,n ∩ I b,n = ∅ which is acontradiction. And that completes the proof.It is clear that for the case of two pseudo-orbits, this theorem is equivalent to that oneproposed by [8]. In this section, we present the lower bound error applied for two case studies, whichexhibit nonlinear dynamics. We select some natural interval extensions that are equivalent.The two chosen models are for the systems sine map and Duffing-Ueda and all routines areperformed in Matlab R2016a in a double precision. We used a computer with a processorDual core @ 2.7GHz and a Windows 8.1 Professional operating system.
A unidimensional sine map is defined as x n +1 = α sin( x n ) , (5)where α = 1 . π . A polynomial NARMAX identified for this system is given by [10] y n +1 = 2 . y n − . y n . (6)Let us consider four equivalent interval extensions of the model 6: F ( X n ) = 2 . X n − . X n , (7) G ( X n ) = 2 . X n − (0 . X n ) X n , (8) H ( X n ) = 2 . X n − . X n X n X n , (9) L ( X n ) = X n (2 . − . X n X n ) . (10)Equations (7)-(10) are mathematically equivalent, but they represent a different sequenceof arithmetic operations. These extensions were simulated using X = 0 . Considering a damped, periodically forced nonlinear Duffing-Ueda oscillator [3]: d ydt + k dydt + µy = A cos( t ) . (11)where µ is the cubic stiffness parameter, k is a linear damping and A is the amplitude ofexcitation. A polynomial NARMAX for the Duffing-Ueda oscillator was identified by [2]. y n +1 = 2 . y n − . y n − + 0 . y n − + 0 . u n + 0 . u n − − . y n + 0 . y n y n − − . y n y n − y n − + 0 . y n − (12) where u = A cos( kT s ), n ∈ N and T s = π/
60. Let us consider four interval extensions ofEq. (12): F ( X n ) = 2 . X n -1 . X n -1 + 0 . X n -2 + 0 . U n + 0 . U n -1 − . X n + 0 . X n X n -1 -0 . X n X n -1 X n -2 + 0 . X n -2 (13) G ( X n ) = 0 . U n + 0 . U n -1 + 2 . X n -1 . X n -1 + 0 . X n -2 − . X n + 0 . X n X n -1 − . X n X n -1 X n -2 + 0 . X n -2 (14) H ( X n ) = 0 . U n + 0 . U n -1 + 2 . X n − . X n -1 + 0 . X n -2 − . X n +0 . X n X n -1 − . X n X n -1 X n -2 + 0 . X n -2 X n -2 X n -2 (15) L ( X n ) = 2 . X n − . X n -1 + 0 . X n -2 + 0 . U n + 0 . U n -1 − . X n X n X n + 0 . X n X n -1 − . X n X n -1 X n -2 + 0 . X n -2 (16) Figure 1(b) shows the evolutions of the maximum lower bound error for the Duffing-Uedaoscillator with the Lyapunov exponent associated. This largest Lyapunov was calculatedby method developed in [6] with value 0.1202 for the maximum of the pseudo-orbits. On-cemore, the computation are in good agreement with the values found in literature, whichfor this model was calculated in 0.115 [2]. n -60-40-20020 l o g ( m a x | ˆ x i , n − ˆ x j , n | ) (a) Sine Map t -40-30-20-10010 l o g ( m a x | ˆ x i , n − ˆ x j , n | ) (b) Duffing-Ueda Oscillator Figure 1: Evolution of the maximum lower bound error.
The errors present in numerical simulations can lead to an erroneous result that doesnot correspond to the real situation of the problem. These errors can be of the repre-sentation of the model, of the insertion of data, of the numerical algorithm, due to thesimplifications, of truncation, of rounding and among others. Thus, some methods havebeen proposed in order to measure these error, but they are complex from the computa-tional point of view and from the mathematical approximation.We presented a method to calculate a lower bound error for free-run simulation ofthe polynomial NARMAX. Our method is based on the comparison of k pseudo-orbits ofthe same models, but derived from different extension intervals. It makes use of recursivefunctions, which increases the relevance of this observation.The methodology was applied in two cases, which are examples of identified systemsobtained from literature, by means of the polynomial NARMAX. The sine map and Duff-ing Ueda oscillator are well known chaotic systems and have been identified using thepolynomial NARMAX.When we compare k pseudo-orbits that are equivalent from the point of view of intervalanalysis, we proved a theorem that the maximum distance of the pseudo-orbits is greaterthan the distance of two pseudo-orbits. This maximum value represents a small differencerespect to the lower bound error for two pseudo-orbits, but reduces the lower bound error.To prove this statement, the Lyapunov exponent was calculated for the maximum lowerbound error, and in both models (sine map and Duffing-Ueda) the result was very closeand in good agreement with values calculated in literature. Acknowledgments
We thank CAPES, CNPq/INERGE, FAPEMIG and UFSJ for their support.