Identification of Nonlinear Dnamic Sstems with Multiple Inputs and Single Output using discrete-time Volterra Tpe Equations Thomas Treichl, Stefan Hofmann, Dierk Schröder Institute for Electrical Drive Sstems Technische Universität München Arcisstraße 2, D 8333 München German Abstract In this article a theoretical framework is presented about the identification of nonlinear dnamic sstems with multiple inputs and single output. Basic considerations about discrete-time Volterra tpe equations are performed. It is shown how measuring equations can be obtained from nonlinear and arbitrar coupled linear time-invariant transfer functions and nonlinear static functions. The principles of sstem identification on fundamental basics of the Volterra theor are presented and with a simulation example the results of the presented approach are evaluated. Introduction Real mechatronic sstems are in general highl complex nonlinear dnamic sstems. A special class of these sstems are Single Input and Single Output SISO) sstems where the term nonlinear describes one or more existing nonlinear static functions. If the structure and the linear parameters of the sstem are well-known then the nonlinear static functions can be identified b Neural Networks within an observer []. In the case that structural knowledge and the knowledge about the linear parameters of the sstem is onl partiall available this identification method fails and other identification algorithms have to be used. Since most of the existing identification methods are derivatives of the linear difference equation stabilit can either not or onl hardl be proven. Furthermore if the disturbed sstem s output of the unknown sstem is fed back to the identification algorithm then the disturbances have negative effects on the identification result [2]. A numerical series to approximate functions with one function variable the so called Volterra series became practicable for the identification of nonlinear dnamic SISO sstems b appling an adaptation law [3]. A detailed knowledge about the structure of the unknown sstem is not necessar but reduces the need for computational performance. Together with the learning algorithm stabilit can be proven. Measuring disturbance influences the adaptation law but not the input of the identification algorithm. Better results can be achieved in comparison to identification techniques based on the difference equation. Identification techniques based on the Volterra theor can be used if the impulse response can be
truncated at a finite point of time. Nevertheless much computational performance would be needed to identif such sstems without an modifications at the identification algorithm. The use of Orthonormal Base Functions OBFs) in order to approximate the truncated impulse response reduces the number of unknown parameters enormousl. With onl little loss of information the truncated impulse response can completel be characterized b an weighted additive overla of OBFs [4]. A more general class are sstems with Multiple Inputs and Single Output MISO) which also include the class of SISO sstems. For those sstems the term nonlinear becomes double-meaning. On the one hand there ma appear nonlinear static functions and on the other hand there also ma appear nonlinear couplings between the multiple inputs within a MISO sstem. In the past ears there have onl been a few investigations about describing MISO sstems on fundamental basics of the Volterra theor. Therefore the identification of nonlinear dnamic sstems with multiple inputs and single output using discrete-time Volterra tpe equations is presented in this article. This article focuses MISO sstems built of nonlinear coupled subsstems. Linear coupled subsstems building a MISO sstem have alread been investigated in [5]. 2 SISO sstem description The discrete-time Volterra equation [3] is displaed in equation 2.) where [k] R is the sstem s output and u[k i] R is the sstem s input at different points of time. g[i] R are the elements of the Volterra kernels and c R is the stead state value. The number of multiplied input variables u[k i] within each Volterra kernel is a characterism for its order, e.g. u[k i ] u[k i 2 ] indicates that the corresponding elements g[i, i 2 ] build a 2 nd order Volterra kernel. [k] = c + + g[i ] u[k i ] + g[i, i 2 ] u[k i ] u[k i 2 ] +... + i = i = i q= i = i 2 = g[i,..., i q ] u[k i ] u[k i q ] 2.) In this article sstems will be considered whose impulse response can be truncated at a finite point of time. Therefore the upper limit of the sums can be substituted b the integer length of the truncated impulse response m N +. Further if just sstems are allowed where the input signal does not directl influence the output [6] then the lower limit of the sums i = can be replaced b i =. Then the discrete-time Volterra equation can be rewritten 2
into a simplified discrete-time Volterra tpe equation m m m [k] = c + g[i ] u[k i ] + g[i, i 2 ] u[k i ] u[k i 2 ] +... + + i q= i 2 = m m g[i,..., i q ] u[k i ] u[k i q ] 2.2) Despite these simplifications the number of Volterra kernel elements in equation 2.2) is still too large for the practical use in the field of sstem identification. Therefore the number of Volterra kernel elements has to be reduced b the introduction of OBFs. With onl little loss of information the Volterra kernels can completel be rebuilt b a weighted additive overla of OBFs. The OBFs used in this article are distorted sinus functions [4] which are defined in equation 2.3). The elements r[j, i] R are orthonormalized b the Cholesk decomposition in order to obtain the orthonormal elements r[j, i] R [7]. r[j, i] = m 2 sin j π e i.5 ζ )) with j m r and i m 2.3) With the form factor ζ R + and the number of distorted sinus functions m r N + the OBFs can be adjusted to the dnamics of the sstem. In figure on the left side orthonormal distorted sinus functions are displaed. B an additive overla of the weighted OBFs the truncated impulse response can be reconstructed, cf. figure on the right side. In literature other OBFs can be found and used instead of orthonormalized distorted sinus functions, e.g. Laguerre [8] or Kautz functions [9]..2..8..6.4.2..2 2 3 4 i.2.4 2 3 4 i Figure : Orthonormal distorted sinus functions left) and the weighted OBFs reconstructing a truncated impulse response dashed) b an additive overla right). If the OBFs are applied to equation 2.2) a reduced number of Volterra kernel elements g r R appears within each Volterra kernel. B inserting the OBFs into equation 2.2) the 3
new Volterra tpe equation can be written as m m r [k] = c + g r [j ] r[j, i ] u[k i ] + + + m i 2 = m i 2 = j = m m r m m r j = j 2 = m m r i q= g r [j, j 2 ] r[j 2, i 2 ] r[j, i ] u[k i ] u[k i 2 ] +... + m r j = j 2 = m r j q= 3 MISO sstem descriptions g r [j,..., j q ] r[j q, i q ] r[j, i ] u[k i ] u[k i q ] Hammerstein and Wiener models are well-known examples for the identification of SISO sstems u using discrete-time Volterra tpe equations. For this reason the following investigations are illustrated b these basic models. The Hammerstein model consists of a nonlinear static function followed b a Linear Time-Invariant LTI) transfer function as displaed in figure 2 on the left side. 2.4) u NL 5 4 3 2 v u F s) NL F s) Figure 2: Hammerstein model left) and five marked points where a second input would ma be applied in a nonlinear manner right). In this article it is assumed that the nonlinear static function can be described b a polnomial of q th order, i.e. q N +. The LTI transfer function will be characterized b the discrete-time convolution of the truncated impulse response and the input. These equations can be written as m vu) = a u + a u +... + a q u q [k] = h[i] v[k i] 3.) B inserting the polnomial into the convolution the mathematical description for a Hammerstein model can be derived [k] = a m i= h[i] + a m i= h[i] u[k i] + a 2 m i= i= h[i] u 2 [k i] +... + a q m i= h[i] u q [k i] 3.2) B a comparison of equation 2.2) with equation 3.2) the following associations can be found. Each Volterra kernel element equals one element in the mathematical description for 4
the Hammerstein model, e.g. g[i ] = a h[i] or g[i,..., i q ] = a q h[i], i = i =... = i q. The Volterra kernels of higher order are just occupied on the main diagonal when describing a Hammerstein model so that the can alwas be displaed in a two-dimensional space. In figure 2 on the right side five points indicate where a second input ma be applied at the Hammerstein model. In the following paragraphs onl the mathematical descriptions for nonlinear coupled sstems will be formulated. For sstem identification these mathematical descriptions have to be transformed into Volterra tpe equations. In detail, different combinations of polnomial coefficients multiplied with one element of the truncated impulse response equals one Volterra kernel element which are needed for sstem identification. If a second input signal ma be applied to the Hammerstein model, according to point 5 in figure 2 on the right side, the past-time values of the two input signals at the same point of time have to be multiplied. The nonlinear combined input signals are fed through the nonlinear static function and the LTI transfer function. The mathematical formulation for r N + input signals is given b m [k] = { h [i ] a + a u [k i ]... u r[k i ] ) +... + } + a q u q [k i ]... u q r[k i ]) 3.3) The first indices of each element in the previous and the following formulations is an indication to which input signal the element belongs, e.g. a q is the q th coefficient of the polnomial which belongs to the first input. A more general case is the one if an input signal or more are applied at point 4 in figure 2 on the right side. For this case a multi-dimensional polnomial has to be used which is defined for r input signals in the following equation vu, u 2,..., u r ) = q q 2 n = n 2 = q r n r= a n n 2...n r u n u n 2 2 u nr r 3.4) Different orders in each direction ma have to be used for describing a multi-dimensional nonlinear static function. Therefore the upper limits of the sums are defined b q... q r N +. After inserting equation 3.4) into equation 3.) the mathematical formulation for this case can be obtained } { m q [k] = h [i ] q 2 n = n 2 = q r n r= a n n 2...n r u n [k i ] u n 2 2 [k i ] u nr r [k i ] 3.5) If an input signal will be applied at point 3 as displaed in figure 2 on the right side it can be shown that this case is alread included in equation 3.5). The past time-values of the input signal u 2 [k i ] will just occur to the power of one. In the case of r input signals applied at the same point all the input signals except u [k i ] occur to the power of one. 5
Appling another input signal at point 2 in figure 2 on the right side is not allowed as long as F s) is a linear sstem because this would violate the linearit principle and therefore this case will be excluded from the investigations. As defined in section 2, appling a second input signal at point in figure 2 on the right side this would directl influence the output signal and therefore this coupling is also excluded from the investigations. At first sight it could be meant that equation 3.5) is the most general case for forward directed and nonlinear coupled Hammerstein models but the general case is the one if two or r Hammerstein models will be multiplied at the output. For r input signals this leads to the following mathematical formulation which generall includes the descriptions before [k] = m m h [i ] + a h [i ] u [k i ] + a 2 h [i ] u 2 [k i ] +... + a m m + a q h [i ] u q [k i ] )... a r m r m r m r + a r2 h r [i r ] u 2 r[k i r ] +... + a rq m r h r [i r ] + a r h r [i r ] u r [k i r ] + h r [i r ] u qr r [k i r ] replacements In figure 3 on the left side the cases of nonlinear coupled Hammerstein models for r = 2 input signals are displaed which have been investigated et. The sstems which can be described ma have one of the dashed couplings displaed. It can also be seen that not all nonlinear couplings of Hammerstein models have been investigated et. There ma also exist sstems with backward directed, nonlinear couplings as displaed in figure 3 on the right side. ) 3.6) u NL v F s) u 2 v NL 2 u 2 2 F 2 s) u NL NL 2 v v 2 c a b F s) F 2 s) Figure 3: Forward directed nonlinear couplings left) and backward directed nonlinear couplings right) between two Hammerstein models. The mathematical formulations for the cases marked with the letters a... c in figure 3 on the right side can be derived in a similar wa as before for nonlinear coupled Hammerstein models in forward direction. Displaing all mathematical formulations for these cases would go beond the scope of this article because the mathematical descriptions for these cases are implemented in the case marked with the letter c in figure 3 on the right side. 6
The description for r Hammerstein models coupled to another input of a Hammerstein model can be derived b the multiplication of the input signal u [k i ] with r descriptions for Hammerstein models as displaed in the following equation m [k] = a m 2 +... + a 2q m r + a rq m 2 + a 2 i 2 = m h [i ] + a i 2 = h [i ] { h 2 [i 2 ] u q 2 2 [k i 2] h r [i r ] u qr r [k i r ] )} u [k i ] ) a r m r a 2 m 2 m +... + a q m 2 h 2 [i 2 ] u 2 [k i 2 ] +... + a 2q a r m r m r h r [i r ] + a r i 2 = i 2 = m 2 h 2 [i 2 ] + a 2 m r h r [i r ] + a r h [i ] { h 2 [i 2 ] u q 2 2 [k i 2] u q [k i ] m r h r [i r ] u r [k i r ] +... + a rq ) i 2 = h 2 [i 2 ] u 2 [k i 2 ] + h r [i r ] u r [k i r ] +... + a 2 m 2 h 2 [i 2 ] + i 2 = h r [i r ] u qr r [k i r ] )} 3.7) The mathematical formulation for a Wiener model can be derived in a similar wa as alread explained for the Hammerstein model. The Wiener model consists of a LTI transfer function followed b a nonlinear static function as displaed in figure 4 on the left side. Inserting a discrete-time convolution into a polnomial leads to a more complex SISO sstem description in comparison to the Hammerstein model. The previous considerations for MISO sstems derived from nonlinear coupled Hammerstein models could also be performed for MISO sstems derived from nonlinear coupled Wiener models but it should be noticed that each MISO sstem derived from nonlinear coupled Wiener models can also be build of nonlinear coupled Hammerstein models. This fact replacements is shown in figure 4 where the sstem depicted on the right side is alread known from the MISO sstem description of nonlinear coupled Hammerstein models and which equals the scheme of a Wiener model displaed in figure 4 on the left side. u v u F s) NL F s) K = NL Figure 4: Wiener model left) as a tpical SISO sstem and the Wiener model consisting of two Hammerstein models in-line right). 7
4 Identification algorithm and stabilit In general the mathematical description of an unknown sstem which can be described with the proposed method and whose impulse response can be truncated at a finite point of time, i.e. SISO sstems [6] as well as linear [5] and nonlinear coupled MISO sstems, can be transformed into the following equation [k] = x T [k] θ + z[k] 4.) θ R pr is the parameter vector including the reduced number of Volterra kernel elements of the sstem and p r N +. The measuring vector x[k] contains m past-time values of the input signal, respectivel for MISO sstems r input signals and special combinations with different powers from the past-time values of the input signal according to the order of the nonlinear static functions and the structure of the sstem so that x[k] R pr. z[k] R is a disturbance which includes the inherent approximation error, quantization noise and measurement noise. The inherent approximation error would disappear if the upper limit of the truncated impulse response would be substituted b infinit. The noise signals will never disappear because ideal output signal measuring will never be possible. The output signal of the identification algorithm ŷ[k] is defined as ŷ[k] = x T [k] θ[k] 4.2) with the unknown parameter vector θ[k] R pr. It can be seen that an equivalent adaptation between an unknown sstem and the identification algorithm will never be possible as long as the disturbance is present. With the difference between the real and the calculated output anpsfrag a-posteriori replacements error signal can be defined of the form e post [k] = [k] ŷ[k] = x T [k] θ θ[k] ) + z[k] = x T [k] φ[k] + z[k] 4.3) Figure 5 on the left side shows the general identification structure with the unknown sstem and the identification algorithm. As the adaptation law for online identification the Recursive u unknown sstem ident. algorithm z e ŷ u unknown sstem parallel model e ŷ Figure 5: Identification structure for the identification of an unknown sstems left) and validation structure for a qualitative assessment of the identification result right). Least SQuare RLSQ) algorithm is used. On the right side of figure 5 the validation structure 8
is displaed. For the validation process it is assumed that the unknown sstem is well-known and therefore the disturbance is neglected. As alread pinpointed in the introduction stabilit can be proven b the use of the Lapunov stabilit criterion []. For the following introduction into the stabilit proof the disturbance will be neglected and set to z[k] =, k. It has to be known that the covariance matrix P [k] of the RLSQ algorithm is smmetrical and positive definite at ever point of time. Therefore the inverse of the covariance matrix P [k] does exist and is also positive definite. The a-priori error signal can be calculated with e[k] = x T [k] θ θ[k ) ] = x T [k] φ[k ] 4.4) This a-priori error signal e[k] is used to adjust the parameters of the parameter vector θ[k]. The mathematical formulation for updating the parameter vector θ[k] and for updating the parameter error vector φ[k] can be written as θ[k] = θ[k ] + P [k ] x[k] e[k] + x T [k] P [k ] x[k] φ[k] = φ[k ] + P [k] x[k] e[k] 4.5) The Lapunov function candidate is defined b V [ φ[k] ] = φ T [k] P [k] φ[k] 4.6) With the knowledge about the covariance matrix P [k] R pr pr and its inverse P [k] and with the equations 4.3), 4.4) and 4.5) it can be shown that with the Lapunov function candidate the following mathematical formulation can be obtained V [ φ[k] ] = V [ φ[k] ] V [ φ[k ] ] e 2 [k] = + x T [k] P [k] x[k] 4.7) The nominator and with x T [k] P [k] x[k], k also the denominator of the previous equation is alwas positive. Therefore the whole fraction is negative at ever point of time. With V [ φ[k] ] V [] the limes towards infinit can be calculated so that lim e 2 [k] k + x T [k] P [k] x[k] = = lim e[k] = 4.8) k In other words it can be shown that the limes of the a-priori error tends to zero and therefore the parameter vector of the identification algorithm must tend to the parameter vector of the unknown sstem if the sstem is persistentl excited and the disturbance is neglected. For this reason the output signal of the identification algorithm is limited at ever point of time if the input signal of the identification algorithm is also limited and if the impulse response of the unknown sstem can be truncated at a finite point of time. 9
u NL v F s) z output ŷ[k] output [k] u 2 NL 2 v 2 F 2 s) MISO model e 2 5 5 2 25 t [s] 5 error e[k] ŷ Identification algorithm 5 2 4 6 8 t [s] Figure 6: Structure of an unknown MISO sstem and identification algorithm left) and the outgoing signals during the identification process right). 5 Identification example In this section the identification of two nonlinear coupled Hammerstein models will be illustrated as depicted in figure 6 on the left side. For the identification of this unknown sstem equation 3.6) has to be transformed into a discrete-time Volterra tpe equation. As alread described for a Hammerstein model in section 3, each combination of a polnomial coefficient multiplied with an element of the truncated impulse response equals an Volterra kernel element. The derived discrete-time Volterra tpe equation can then be rewritten in form of a measuring equation which can finall be used for the identification of the unknown sstem. The LTI transfer functions and the nonlinear static functions of the two Hammerstein models are given b F s) =.6 +.8s + s 2 v u ) =..3u +.5u 2 F 2 s) = 2.6 +.4s + s 2 v 2 u 2 ) =.3.7u 2 u 2 2 To the sstem s output of the unknown sstem a white Gaussian noise signal is added. The identification parameters are chosen as follows r = 2 h =.s q = 2 q 2 = 2 m 2 h = 4 ζ 2 h = 39 m r = 8 m h = 35 ζ h = 63.6 m r = 6 N h = 25 with h R + as the sampling time and N N + the value for the identification period of the identification process and the validation process. Without the use of OBFs p = 646 parameters would have to be adapted but with the use of OBFs the reduced number of
unknown parameters is p r = 22. In figure 6 on the right side the outgoing signals during the identification process are displaed. At the top of this figure the output signal [k] and the calculated output signal ŷ[k] are displaed and at the bottom the a-priori error signal e[k] is depicted..9.8.7.6.5.4.3.2..2 3 5 5 2 25 3 i 2 i.8.6.4.2 2 i 2 3 Figure 7: Linear Volterra kernels occur if linear parts are present left) and two-dimensional smmetrical Volterra kernel occur at special MISO structures right). Tpical examples of Volterra kernels are displaed in figure 7 and figure 8 on the left side. The linear Volterra kernel alwas appears if a linear part in a SISO sstem as well as MISO.6.4 5 5 output ŷ[k] output [k].2 2 5 5 2 25 t [s] 2 error e[k].2 3 2 i 2 i 2 4 5 5 2 25 t [s] Figure 8: Two-dimensional unsmmetrical Volterra kernel occur at special MISO sstems left) and the outgoing signals during the validation process right).
sstem does exist. Two-dimensional smmetrical Volterra kernels occur in MISO sstems if the outputs of two LTI transfer functions are multiplied and F s) = F 2 s) or if a LTI transfer function is followed b a nonlinear static function, i.e. Wiener model structures. The two-dimensional unsmmetrical Volterra kernel is the result of the multiplication of two output signals from two different LTI transfer functions, i.e. F s) F 2 s). The result of the undisturbed validation process is depicted in figure 8 on the right side where the output signal of the unknown sstem [k], the output signal of the parallel model ŷ[k] and the a-priori error signal e[k] are depicted. At the beginning of the validation process an error is visible because the measuring vector x[k] is filled with zeros and therefore a wrong output signal within the parallel model ŷ[k] is calculated. If the measuring vector is completel filled with past-time values of the input signal the output error tends near zero. 6 Conclusion and future aspects In this article a new approach for the identification of nonlinear dnamic sstems with multiple inputs and single output using discrete-time Volterra tpe equations has been presented. It has been shown that within MISO sstems the term nonlinear becomes double-meaning, i.e. nonlinear static functions as well as nonlinear couplings within a MISO sstem ma exist. Basic considerations about the Volterra theor have been performed and the mathematical descriptions for different nonlinear coupled MISO sstems have been presented. The identification structure as well as basic considerations about the stabilit proof have been shown. Finall an identification example of nonlinear coupled Hammerstein models and its arising results have been displaed. Further work has to be done in code optimization due to the fact that still a high number of parameters is needed and therefore a lot of computational performance. The unification of the proposed sstem identification method with the existing observer theor is planned for the observation of globall integrating MISO sstems []. Furthermore the developed basics will be used for sstem identifications in the field of mechatronic applications. 2
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