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Mathematical and Computer Modelling of Dynamical Systems
, Volume 14 (3): 18 – Jun 1, 2008

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- Taylor & Francis
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- Copyright Taylor & Francis Group, LLC
- ISSN
- 1744-5051
- eISSN
- 1387-3954
- DOI
- 10.1080/13873950701844873
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Mathematical and Computer Modelling of Dynamical Systems Vol. 14, No. 3, June 2008, 213–230 Model inversion of boundary controlled parabolic partial diﬀerential equations using summability methods a b T. Meurer * and M. Zeitz a b Automation and Control Institute, Vienna University of Technology, 1040, Vienna, Austria; Institute for System Dynamics, Universita¨t Stuttgart, 70569, Stuttgart, Germany (Received 4 December 2006; ﬁnal version received 15 September 2007) The combination of formal power series and appropriate summability methods is considered for the inversion of the non-linear, distributed-parameter model of a boundary controlled tubular reactor. The inversion is performed in order to realize the tracking of certain prescribed output trajectories in open-loop control. Simulation results illustrate the applicability of the design approach for the example of ﬁnite-time transitions between set-points for a tubular bioreactor. Keywords: model inversion; distributed-parameter system; diﬀerential ﬂatness; formal power series; summability methods; feedforward tracking control; trajectory planning; tubular reactor 1. Introduction First principles modeling of chemical processes leads to a distributed-parameter description in terms of (non-linear) partial diﬀerential equations (PDEs) whenever spatial or dispersed-phase eﬀects have to be taken into account. Well-known examples concern (bio-)chemical tubular and ﬁxed-bed reactors for production or degradation with complex dynamical behaviour and multiple steady states [1,2]. From a control perspective, it can be observed that model-based control design methods for distributed-parameter systems (DPSs) are mainly restricted to the linear case and directed towards the solution of the stabilization problem (see, e.g. [3] and the references therein). However, for chemical process operation it is required to startup a reactor, realize transitions between set-points, and ﬁnally shutdown a reactor. Besides the stabilization of the set-points, these requirements inherently constitute a tracking control problem for the DPS so that its outputs track certain predeﬁned reference signals. For this, besides optimal control for PDEs [4–7] with it computational drawbacks, only a few analytical approaches exist. Considering tracking control problems for ﬁnite-dimensional non-linear systems, diﬀerential ﬂatness is a well established tool for inversion-based system analysis, trajectory planning, feedforward, and feedback tracking control design [8–10]. The ﬂatness property implies that there exist so-called ﬂat or basic outputs, which allow to parametrize system *Corresponding author. Email: meurer@acin.tuwien.ac.at Revised and expanded version of a paper presented at the 5th Vienna International Conference on Mathematical Modelling (MATHMOD), Vienna, Austria, 2006. ISSN 1387-3954 print/ISSN 1744-5051 online 2008 Taylor & Francis DOI: 10.1080/13873950701844873 http://www.informaworld.com 214 T. Meurer and M. Zeitz states and inputs in terms of the basic outputs and its time-derivatives up to a certain problem dependent order. Recent extensions consider the application of ﬂatness-based methods to solve tracking control problems for linear and non-linear parabolic as well as hyperbolic distributed-parameter systems (see, e.g. [11–16] and the references therein). For this, inversion-based feedforward tracking control is systematically determined as is schematically outlined in Figure 1: the input trajectory u*(t), which is needed to track a desired trajectory y*(t) pre-scribed from the signal generator S*, is obtained from the inverse model S of the DPS S . In addition, also the respective state proﬁle x*(z, t) can be determined in terms of y*(t), which provides further insight into the system dynamics and allows to implicitly incorporate possible state and input constraints into the trajectory planning by suitably modifying y*(t). It is obvious that the rigorous determination of S requires the inversion of the inﬁnite-dimensional model S . For a certain class of non-linear parabolic DPSs with boundary input, the inversion can be performed explicitly using the formal power series (FPS) [12–14,16]. Thereby, the state variables are assumed to follow the power series in the spatial coordinate with time- varying coeﬃcients to be determined from the diﬀerential recursion obtained after substitution of the FPS into the governing PDEs and boundary conditions (BCs). The solution of the diﬀerential recursion can be obtained by introducing a basic output, which, in accordance with the ﬂatness approach for ﬁnite-dimensional systems, allows to diﬀerentially parametrize the series coeﬃcients and hence the state variables and boundary inputs via the power series. However, the applicability of the formal power series solution is in general rather limited due to the required proof of uniform series convergence, which is related to the problem of trajectory planning [13]. In the following, the application of FPS is considered for the inversion of a simpliﬁed tubular reactor model given in terms of a scalar non-linear parabolic PDE. As is illustrated in Section 2, a basic output parametrizing the state and the boundary input can be determined. The respective proof of convergence yields rather restrictive conditions, which relate system parameters and trajectory planning. To overcome this problem, so-called summability methods [17,18] to sum slowly converging as well as certain divergent series are considered in Section 3. Thereby, a novel approximate summation method is introduced in view of the application to parametrized FPS resulting from DPS control problems, where, due to the increasing complexity arising from the non-linear terms, only a ﬁnite number of parametrized series coeﬃcients can be computed. To provide an eﬃcient symbolic computation of a suﬃciently large number of series coeﬃcients, an algorithm is proposed in Section 4, which can be easily integrated in computer-algebra-systems such as MATHEMATICA or MAPLE. The applicability of the proposed approach is validated by simulation results for feedforward tracking control for the considered tubular reactor in Section 5. Some ﬁnal remarks conclude the paper. 2. Model inversion using formal power series To illustrate the model inversion approach and to demonstrate the emerging drawbacks which appear within the classical framework of uniformly convergent power series, the Figure 1. Inversion-based feedforward control for DPS S with inverse model S and signal generator S*. Mathematical and Computer Modelling of Dynamical Systems 215 example of an anaerobic digestion process in a tubular bioreactor following the reaction scheme X ! B þ gas, with substrate X (to be degraded) and biomass B is considered [19]. In line with physical evidence, the biomass concentration is only slowly varying compared with the substrate concentration and is hence assumed constant, i.e. B(z, t) ¼ B. For the following computations, a dimensionless model is used in the normalized concentration x(z, t) ¼ X(z, t)/K with K the saturation constant such that the normalized substrate x x concentration x(z, t) satisﬁes x(z, t) 2 [0, 1]. Furthermore, the dimensionless reaction kinetics m(x(z, t),B) is assumed to follow the Monod law [19], i.e. b ðBÞxðz; tÞ mðÞ xðz; tÞ; B¼ ð1Þ b ðBÞþ xðz; tÞ with b (B) 4 0, j ¼ 0, 1. In the subsequent analysis, the reaction rate m(x(z, t), B)is approximated by a third order polynomial mðÞ xðz; tÞ; B p x ðz; tÞ within the j¼1 practically interesting operation range 0 5 x(z, t) 5 1 of sub-saturation [20]. The respec- tive dimensionless diﬀusion-convection-reaction model with Danckwert’s BCs reads as @xðz; tÞ 1 @ xðz; tÞ @xðz; tÞ S : ¼ þ þ p x ðz; tÞ; z 2ð0; 1Þ; t > 0 ð2Þ 1 j @t Pe @z @z j¼1 @x 1 @x ð0; tÞ¼ 0; ð1; tÞ¼ uðtÞ xð1; tÞ; t > 0 ð3Þ @z Pe @z yðtÞ¼ xð0; tÞ; t 0: ð4Þ Here, u(t) denotes the boundary input, y(t) the controlled output, and xðz; 0Þ¼ x ðzÞ; z 2½0; 1 represents the initial condition which is assumed to correspond to a stationary solution of (2)–(4). The parameter Pe denotes the Peclet number relating the 2 2 convective (*@x/@z) and diﬀusive (*@ x/@z ) eﬀects. Note that more complex examples in several state variables with Arrhenius-type reaction kinetics resulting from mass and energy balancing in the non-isothermal case can be considered similarly [16,21]. 2.1. Formal power series parametrization System (2)–(4) serves as the basis to illustrate the inversion approach. For this, assume that x(z, t) can be represented by a formal power series, i.e. xðz; tÞ! xðz; tÞ where x ^ðz; tÞ¼ x ^ ðtÞz : ð5Þ n¼0 Note that the term formal denotes the fact that the radius of convergence of the series (5) might be equal to zero. The substitution of x ^ðz; tÞ into the PDE (2), the homogeneous BC (3) at z ¼ 0 and the output (4) yields that hi Pe ^ ^ ^ ^ x ðtÞ¼ x ðtÞðn þ 1Þx ðtÞ cðÞ x ðtÞ ; n 0 nþ2 n nþ1 n n ðn þ 2Þðn þ 1Þ ð6Þ x ^ ðtÞ¼ 0; x ^ ðtÞ¼ yðtÞ; 1 0 216 T. Meurer and M. Zeitz where the reactive term is evaluated using Cauchy’s product formula, i.e. "# n i X X ^ ^ ^ ^ ^ ^ cðÞ x ðtÞ¼ p x ðtÞþ x ðtÞ p x ðtÞþ p x ðtÞx ðtÞ ð7Þ 1 n ni 2 i 3 j ij n n i¼0 j¼0 with x ^ ¼½x ^ ; x ^ ; ... ; x ^ . Obviously, the inﬁnite set of equations (6) for n 0 can be 0 1 n interpreted as a (diﬀerential) recursion for the series coeﬃcients x ^ ðtÞ, n 2 with the starting conditions for x ^ ðtÞ and x ^ ðtÞ. Hence, sequential evaluation provides the result 0 1 that the state x(z, t) and the input u(t) can be parametrized in terms of yðtÞ¼ x ^ ðtÞ¼ xð0; tÞ and its time-derivatives up to inﬁnite order, i.e. 1 1 X X 1 n S : x ^ðz; tÞ¼ x ^ y ðtÞ z ; uðtÞ¼ u ^ ðtÞ; ð8Þ n n n¼0 n¼0 ðn Þ ^ ^ ^ where u ðtÞ¼ x y ðtÞ þðn þ 1Þ=Pex y ðtÞ and y ðtÞ¼ yðtÞ; .. . ; y ðtÞ with n *¼ n n nþ1 n n n (n – nmod2)/2. This in particular constitutes the inverse model S . As outlined above, appropriately assigning a smooth desired trajectory t 7! y ðtÞ2 C for y(t) immediately yields the corresponding input trajectory u(t) ! u*(t) which at ﬁrst is needed to track y*(t) in open-loop and at second realizes the corresponding transient state proﬁle x(z, t) ! x*(z, t). However, this requires to ensure uniform convergence of the series x ^ðz; tÞ with coeﬃcients x ^ ðtÞ, n 0 parametrized in terms of y*(t). Remark 1: It is interesting to note that it is not necessarily required to approximate the considered Monod kinetics (1) by a polynomial in order to apply the formal power series approach. Alternatively, consider b ðBÞx ^ðz; tÞ n 0 ^ ^ mðxðz; tÞ; BÞ¼ m ðtÞz ¼ b ðBÞþ x ^ðz; tÞ n¼0 such that due to Cauchy’s product formula the coeﬃcients m ^ ðtÞ; n 2 N follow from the recursion b ðBÞx ^ ðtÞ x ^ ðtÞm ^ ðtÞ n j 0 nj j¼0 m ^ ðtÞ¼ ; n 1 ð9Þ b ðBÞþ x ^ ðtÞ b ðBÞx ^ ðtÞ m ^ ðtÞ¼ : ð10Þ b ðBÞþ x ðtÞ Diﬀering from the analysis above, this approach requires to solve a system of recursions consisting of (9) and (6) with cðÞ x ^ ðtÞ replaced by m ^ ðtÞ. The tools to be developed in the n n n subsequent sections can also be directly applied in this case. However, since it is to the knowledge of the authors so far not possible to rigorously address the respective problem of uniform series convergence, a detailed analysis of the resulting state and input parametrizations is not pursued within this work. Mathematical and Computer Modelling of Dynamical Systems 217 2.2. Convergence and trajectory planning To obtain a meaningful solution from the formal power series and to validate the mathematical operations such as series diﬀerentiation, interchange of summation, and application of Cauchy’s product formula, uniform convergence of the formal solution (8) has to be veriﬁed with at least a unit radius of convergence since z 2 [0, 1]. For the veriﬁcation of series convergence, the notion of a Gevrey class is needed [22]. The function y*(t) is in the Gevrey class of order a in O R ,if y ðtÞ2C ðOÞ and for every closed subset S of O, there exist positive constants M and R such that for all k 2 N and t 2 S ðkÞ sup y ðtÞ ðk!Þ : ð11Þ t2S Furthermore, y*(t) is either entire, analytic or non-analytic if a 5 1, a ¼ 1or a 4 1, respectively. In particular, uniform convergence of (8) with at least a unit radius of convergence can be ensured if y*(t) is in the Gevrey class of order a 2 while in addition stringent conditions on the system parameters Pe and p , j ¼ 1, 2, 3 have to be imposed, which are typically not met in applications [13,16,21], i.e. sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 1 1 4 1 þ 1 þ þðÞ jp jþ Mjp jþ M jp j : ð12Þ 1 2 3 Pe R 2 Pe The proof of these conditions utilizes some ideas of [13] and extends these to the case of cubic nonlinearities [16]. In particular, it can be veriﬁed by induction that the series coeﬃcients determined by (6) satisfy MEðÞ ðn þ kÞ! ðkÞ sup x ðtÞ ; E > 0: n a R ðn!Þ t2S In this case, the Cauchy-Hadamard formula implies that the series (5) has a radius of convergence equal to 1/E given y(t) of Gevrey order a 2 [1, 2]. After some tedious computations, an estimate of the form MEðÞ ðn þ kÞ! ðkÞ sup x ^ ðtÞ hðEÞ; R ðn!Þ t2S Pe 1 E þjp jþjp jM þjp jM 0 1 2 hðEÞ¼ þ E R 2 is obtained. Solving h(E) ¼ 1 for E and taking the maximal E together with the requirement that E 1 yields condition (12). Note that in the special case p ¼ 0, j ¼ 2, 3, i.e. (2)–(4) represents a linear system, it suﬃces to require that a 2 without further conditions to ensure uniform convergence of (8). 218 T. Meurer and M. Zeitz Hence, except for the linear case, the proof of convergence relates the parameters of the considered system Pe, p , j ¼ 1, 2, 3 with those of the desired trajectory M and R from the respective Gevrey class (11). To illustrate the consequences of the inequality (12) consider Figure 2. There, the admissible domains are depicted in two diﬀerent parameter spaces: (a) for ﬁxed Pe, M, and R with p , j ¼ 1, 2, 3 varied, and (b) for ﬁxed p , j ¼ 1, 2, 3 with Pe, M, and R varied. It can be easily deduced that uniform convergence can be only ensured for rather restrictive parameter sets. In particular the domain depicted in Figure 2(a) contracts to the origin for Pe ¼ 2/3 and vanishes afterwards. On the other hand as is illustrated in Figure 2(b), for a given set of reaction parameters, the possible amplitudes for the desired trajectories y*(t) in terms of the parameters M, R from the Gevrey estimate decrease with increasing Peclet-number Pe. Hence a direct application of the FPS approach is only possible for diﬀusion dominated systems characterized by very low Pe values. Note that this is in strong contrast to the technical requirements where tubular reactors are typically operated at medium or high Pe values. A further drawback arises from the fact that in general only a ﬁnite number of series coeﬃcients x ðtÞ, n ¼ 0,..., N can be determined due to the complexity increasing in n when evaluating the recursion (6) to obtain the coeﬃcients x ^ ðtÞ. However, although convergence can be ensured by satisfying the rather restrictive condition (6), no estimate on the respective speed of convergence of the parametrized power series can be deduced. Hence, it can be observed that the computational limit imposed on the computable number of parametrized series coeﬃcients might not necessarily be suﬃcient to obtain an accurate approximation of the respective series limit. To illustrate this, in the sequel the transition from an initial stationary operating proﬁle x ðzÞ to a ﬁnal stationary operating proﬁle x ðzÞ in ﬁnite time t 2 [0, T], T 4 0 is considered. This is a common task for the startup, the operation, or the shutdown of a reactor. Due to the possibility to parametrize the state x(z, t) and the input u(t) in terms of the output y(t) and its Figure 2. Regions of uniform convergence due to (12) for a variation of the reaction parameters p , j ¼ 1,2,3 with Pe, M, and R ﬁxed in (a) and for a variation of Pe, M, R with p , j ¼ 1,2,3 ﬁxed in (b). Convergence is preserved for parameter values inside the body in case (a) and for parameters enclosed by the depicted surface and the coordinate axes in case (b). Mathematical and Computer Modelling of Dynamical Systems 219 time-derivatives up to inﬁnite order, stationary proﬁles x (z) can be characterized as solutions to the following boundary-value problem deduced from (2)–(4) 2 3 1 d x ðzÞ dx ðzÞ S S j 0 ¼ þ þ p x ðzÞ; z 2ð0; 1Þð13Þ Pe dz dz j¼1 dx ð0Þ¼ 0; x ð0Þ¼ y ; ð14Þ S S dz where y denotes a stationary output value. The respective input u ¼ dx =dzð1Þþ x ð1Þ S S S S Pe follows from the inhomogeneous BC (3) at z ¼ 1. Obviously, re-planning y results in diﬀerent stationary proﬁles x (z). This property, which directly carries over the deduced inverse model representation (8) to the stationary case, can be exploited for trajectory planning, i.e. the assignment of a suitable y ðtÞ2C . Therefore, choose 0 T 0 y ðtÞ¼ y þðy y Þ Y ðtÞð15Þ s;T S S S where 0if t 0 >1if t T t=T F ðtÞdt Y ðtÞ¼ s s;T if t 2ð0; TÞ > R F ðtÞdt with exp s if t 2ð0; 1Þ ðÞ ð1tÞt F ðtÞ¼ 0if t2ð = 0; 1Þ. The trajectory y*(t) is non-analytic at t 2 {0,T} and its Gevrey order can be determined as a ¼ 1 þ 1/s [11,13]. Due to the non-analytic property, it holds that y ðt Þ¼ t ðkÞ y ; y ðt Þ¼ 0 for t* 2 {0,T} and k 1. This allows to exactly start at the initial 0 0 stationary proﬁle x ðzÞ determined from (13), (14) with y ¼ y and exactly reach the ﬁnal S S T T stationary proﬁle x ðzÞ obtained similarly with y ¼ y . Note that a solution to the S S transition problem is not possible using an analytic function due to Liouville’s theorem [23]. As outlined above, for p ¼ 0, j ¼ 2, 3, uniform convergence of (8) can be ensured provided that the desired trajectory y*(t) for the basic output y(t) is of Gevrey order a 2. The respective radius of convergence is inﬁnite if a 5 2. To illustrate the respective speed of convergence, consider Figure 3. The numerical results are shown for parameters p ¼ 0, j ¼ 1, 2, 3 when truncating the input parametrization at diﬀerent N 2 {10, 20, 30}, i.e. u ðtÞ! u ðtÞ¼ u ^ ðtÞ. For the evaluation of the corresponding tracking behaviour, N n¼0 220 T. Meurer and M. Zeitz Figure 3. Comparison of numerical results for open-loop boundary control of (2)–(4) with p ¼ 0, j ¼ 1, 2, 3 and Pe 2 {5, 10} for N 2 {10, 20, 30} varied. The desired output trajectory (15) is 0 T parametrized with y ¼ 1:0; y ¼ 0:1, T ¼ 4, and s ¼ 2.0. Left: input u ðtÞ; right: output trajectory S S N y(t) ¼ x(0, t) from compared with desired value y*(t). i.e. y(t) ! y*(t) which is equivalent to realize the desired transition between steady states, the obtained open-loop boundary control u ðtÞ is applied to a ‘method-of-lines’ discretized simulation model of the tubular reactor (2)–(4). It can be observed that an increase in the Pe-number, corresponding to a decrease in diﬀusion, requires an increase in Mathematical and Computer Modelling of Dynamical Systems 221 the number N of considered coeﬃcients uˆ (t) to obtain a suﬃciently accurate approximation of the series limit u*(t) by the truncated series u ðtÞ. Note that this is a quite common behaviour when studying the inﬂuence of convective terms on the con- vergence behaviour of the power series parametrization in both the linear as well as non- linear setup [16,24]. However, as outlined above, there is a fast increase in the respective computational eﬀort to symbolically compute the parametrized series coeﬃcients from the diﬀerential recursion (6) for increasing n, which illustrates the necessity to introduce appropriate algorithms for an eﬃcient symbolic recursive evaluation. To overcome the rather stringent convergence conditions such as (12) obtained when studying non-linear PDEs as well as the requirement of an eﬃcient recursive coeﬃcient evaluation, it is shown in the sequel that the combination of formal power series, appropriate summability methods, and suitable algorithms for the symbolic evaluation of diﬀerential recursions provides a general and broadly applicable framework for enhanced motion planning and feedforward tracking control design. 3. Summability methods for power series In the previous section, formal power series are applied to determine the inverse system representation of the boundary controlled parabolic PDE (2)–(4) in terms of the basic output and its time-derivatives up to inﬁnite order. It is shown that convergence of the parametrized power series (8) greatly depends on appropriate trajectory planning, the structure of the DPS, and the respective system parameters, which signiﬁcantly restricts the applicability of the method. One approach to overcome these convergence limitations is provided by the so-called summability methods with the desire to prolong the space of the uniformly convergent power series in order to deal with certain divergent series [17,18,25–28]. Within this framework, general results are mainly available for formal solutions to ordinary diﬀerential equations (ODEs) [18,26–28], with recent extensions to PDEs [29–31]. As the most common example of a summation method consider partial summation as the limit of the partial sum tends to inﬁnity, i.e. 1 N X X n n S x ^ z ¼ lim x ^ z ; z 2 C: ð16Þ P n n N!1 n¼0 n¼0 Hence S can be interpreted as a linear functional mapping each convergent power series to its natural sum. This method is weak in the sense that it applies to convergent series only but provides insight into the structure of summation methods. Generalizing this approach, let S be a linear functional on a linear space X of sequences or equivalent series. For the V S application of S to a formal power series x ^ðzÞ introduce a sequence of functions a (x)in V n the continuous parameter x 0 such that 1 1 X X n n A S x ^ z ¼ lim a ðxÞx ^ z ¼ lim x ^ ðz; xÞ; ð17Þ V n n n x!1 x!1 n¼0 n¼0 A A where x ^ ðz; xÞ abbreviates the weighted sum. If x ^ ðz; xÞ converges for jzj 5 r, it deﬁnes a family of holomorphic functions in a disc D ¼ D(0,r) of radius r centred at the origin. 222 T. Meurer and M. Zeitz Following [32] the restriction to the case where convergence for x ! ? is locally uniform in z (at least for z in some sectorial region O of D) yields a holomorphic function x(z) ¼ lim x (z;x)on O. In this case, the formal power series x ^ðzÞ is said to be x ! ? A-summable on the region O and the function xðzÞ¼ðS x ^ÞðzÞ is referred to as the A-sum of x ^ðzÞ. As a result, the space X consists of all A-summable series and is called the summability domain of the functional S . For a comprehensive discussion, the reader is referred to [17]. In the sequel, let E [[z]] denote the set of all formal power series x ^ðzÞ¼ x ^ z , n¼0 z 2 C with coeﬃcients x ^ 2 E, n 0 in some Banach space E. Furthermore, let A(O,E) denote the set of all functions x(z): O ! E, holomorphic in a sectorial region O C and having x ^ðzÞ as an asymptotic expansion. To make a summation method suitable for the formal solution of ODEs, the method has to satisfy certain requirements [18,32]: (i) The summability domain X has to be a diﬀerential algebra, prolonging the space of convergent power series xðzÞ in z 2 C. (ii) The functional S of (17) has to be regular, summing each convergent power series x ^ðzÞ to its natural sum. (iii) The functional S of (17) has to be a linear homomorphism mapping products to products and derivatives to derivatives. (iv) The operator J: A(O, E) ! E [[z]] has to invert the functional S . Note that in view of the application to PDEs also uniform summability has to be introduced. Property (i) ensures ﬁrst that diﬀerentiation is allowed and second that the summability domain enlarges the set of convergent series. Due to property (ii), the sum (the natural limit) of a convergent series is preserved under the functional S . Assuming that (iii) is satisﬁed, it follows that given x ^ðzÞ; w ^ðzÞ2 X ; SðÞ x ^ðzÞ w ^ðzÞ¼ S V xðzÞ wðzÞ where and denote multiplication in E [[z]] and E, respectively, and S ðdx ^ðzÞ=dzÞ¼ dxðzÞ=dz. Both properties are essential for the study of formal power series solutions to diﬀerential equations and are rarely satisﬁed for general summation processes. Property (iv) basically ensures, that given a formal power series x ^ðzÞ satisfying a given diﬀerential equation, its sum xðzÞ¼ðS x ^ÞðzÞ can be uniquely determined in the region O and solves the diﬀerential equation with x(z) being asymptotic to x ^ðzÞ. One such summation method is the so-called k-summation, which is introduced in [26] as a generalization of the work of [25] on the summation of divergent series. For its deﬁnition, consider ﬁrst the formal Borel operator B of order k of the formal power series xðzÞ2 E½ ½z , i.e. X n x ^ ðzÞ¼ðB x ^ÞðzÞ¼ x ^ : ð18Þ B;k k n ð1 þ Þ n¼0 Assuming that x ^ ðzÞ converges in a neighbourhood of the origin z ¼ 0 and yields an B;k analytical continuation into a sector S(d, E) of the complex domain of bisecting direction d and opening E 4 0 with exponential growth at most k in S(d, E), then xðzÞ¼ðL x ^ ÞðzÞ k B;k \ Mathematical and Computer Modelling of Dynamical Systems 223 denotes the k-sum of the series x ^ðzÞ where L denotes the Laplace-integral of order k [18], i.e. 1ðtÞ ðs=zÞ k1 ðL x ^ ÞðzÞ¼ x ^ ðsÞe s ds: ð19Þ k B;k B;k For details on the domain of convergence of (19) consult [18]. In particular it can be proven that the space of all k-summable power series (in direction d) remarkably has the same algebraic properties as the space of convergent power series. As an illustrative example consider the k-sum of the Euler series, which is divergent for any z 2 C except z ¼ 0 [17, p. 26]: Example 3.1 (k-summation of the Euler series [16]) Consider ﬁrst the application n n of the formal Borel operator B of order k (18) on the series xðzÞ¼ ð1 þ Þz with n¼0 z 2 C, i.e. x ^ ðzÞ¼ðB x ^ÞðzÞ¼ z : B;k k n¼0 Since ðS x ^ ÞðzÞ¼ 1=ð1 zÞ, the formal Borel operator of order k can be analytically P B;k continued into any sector excluding the positive real axis with exponential growth obviously less than k. As a result, the series x ^ðzÞ is k-summable in any direction d 6¼ 2jp, j 2 Z. Hence let k ¼ 1 and replace z by 7z such that x ^ðzÞ corresponds to the Euler series. Since now ðSx ^ ÞðzÞ¼ 1=ð1 þ zÞ, it follows similarly that the Euler series is 1-summable in B;k any direction d 6¼ (2j þ 1)p, j 2 Z. In particular, 1ðtÞ Z Z 1 1 1 s z¼z=ð1þsÞ e e xðzÞ¼ðL x Þ¼ e ds dz; 1 B;1 z 1 þ s z z 0 0 which directly corresponds to Euler’s result. Summability for k 4 1 follows directly from [18, Lemma 9, p. 101]. Alternatively, k-summation can be re-deﬁned as a subset of the means of integral functions approach [17], i.e. P n s ðzÞ ð1þ Þ n¼0 xðzÞﬃ x ^ðzÞ¼ lim ¼: ðS x ^ÞðzÞð20Þ 1 B P n x!1 ð1þ Þ n¼0 as is outlined in [33]. Here, s ðzÞ¼ x ^ z denotes the n-th partial sum. Note that the n n j¼0 denominator function of the fractional term is also known as the Mittag-Leﬄer function of index 1/k, i.e. E ðxÞ¼ x =ð1 þ n=kÞ. As a simple example for the application of 1=k n¼0 S , consider the geometric series. B 224 T. Meurer and M. Zeitz Example 3.2 (Summation of the geometric series by S ) It can be easily veriﬁed that the n-th nþ1 partial sum s (z) of the geometric series x ^ðzÞ¼ z is given by s (z) ¼ (z –1)/(z–1). n n n¼0 Hence it follows that 1 n nþ1 E ðzxÞ z 1 x 1 z 1 1=k k 1 ðS x ^ÞðzÞ¼ lim E ðxÞ ¼ lim ¼ ¼ xðzÞ B 1=k x!1 z 1 ð1 þ Þ 1 z 1 z x!1 E ðxÞ 1 z 1=k n¼0 k provided that Re(z) 5 1. Obviously, the method S allows to sum the geometric series in the whole complex half plane Re(z) 5 1. In view of the application of k-summation to formal power series solutions arising from DPS control problems, the previously discussed theoretically appealing approaches have to be suitably modiﬁed in order to deal with only a ﬁnite number of series coeﬃcients, which leads to so-called generalized sequence transformations [34,35]. These methods allow to accelerate convergence and approximate the sum of certain divergent series using only partial information on the series under consideration and in particular play a crucial role e.g. in quantum physics and quantum chemistry [35]. Within this framework, a modiﬁcation of the previously discussed k-summation in the form (20), namely the so-called (N,x)-approximate k-summation is proposed [16,24], i.e. N n P P x ^ z ð1þn=kÞ n¼0 j¼0 N;x S ðÞ x ^ðz; tÞ¼ ; ð21Þ ð1þn=kÞ n¼0 which can be applied to extract the limit (in the sense of the summability method) of the slowly converging or possibly diverging series (8) from a ﬁnite number of series co- eﬃcients. This is exemplarily studied in the following example for the geometric series. Example 3.3 ((N,x)-approximate k-sum of the geometric series) In Example 3.2, it is shown that the geometric series is k-summable to the classical limit x(z) ¼ 1/(1–z) for any z 2 C with Re(z) 5 1. Taking this as a benchmark, it can be shown that x(z) can be accurately N N;x recovered from a ﬁnite number of series coeﬃcients fg z ; N 2 N by applying S as n¼0 deﬁned in (21) with suitable summation parameters x and k. Table 1 summarizes these Table 1. Numerical evaluation of the (N, x)-approximate k-sum of the geometric series. N;x N x S x ^ðzÞ e(z) s (z) 4 0.8926884 0.41085 0.0775168 11 10 1.7421111 0.33899 0.005656 683 20 3.15724 0.333412 0.00007893 699051 710 14 50 7.37305 0.333333 2.5 6 10 7.5 6 10 k-sum x(z) 3 Mathematical and Computer Modelling of Dynamical Systems 225 results for z ¼ 72 when varying the number N of series coeﬃcients. For ﬁxed k ¼ 1 and increasing N 2 {4,10, 20, 50}, a suitable choice of x yields highly accurate results with N;x absolute error eðzÞ¼ xðzÞðS x ^ÞðzÞ approaching zero. The respective value s (z)of the N-th partial sum is provided to illustrate the divergent (in the classical sense) behaviour of the geometric series for z outside the complex unit circle. In [16], further examples, a discussion on the suitable choice of the summation parameters x, k, and a detailed study of summation techniques in view of tracking control design for parabolic DPS can be found. Furthermore, it should be pointed out, that initial results on the application of divergent series to trajectory planning for the linear heat equation have been presented in [12] where the so-called least-term summation has been applied. Least-term summation at ﬁrst determines the absolutely smallest coeﬃcient of the series – here (8) – and then calculates the partial sum up to this lt smallest term, i.e. u ^ ðtÞ with N from max ju ^ ðtÞj ¼ min max ju ^ ðtÞj. Note that n lt t N n t n n¼0 lt least-term summation is neither regular nor induces the structure of a diﬀerential algebra. In addition, since in general no explicit formula can be obtained for the formal solution series, it is impossible to determine the smallest term. Nevertheless, the results obtained by least-term summation – when the smallest available term is considered to be the smallest overall coeﬃcient – are often quite remarkable. A comparison of the application of partial summation, least term summation, and k-summation is presented in [24]. 4. An algorithm for the eﬃcient evaluation of non-linear diﬀerential recurrence relations As illustrated above, the incorporation of appropriate summability methods allows to greatly enhance the applicability of FPS in view of model inversion and the solution of tracking control problems for boundary-controlled DPS. However, this requires an eﬃcient symbolic computation of the parametrized series coeﬃcients from the diﬀerential recursion (6). For this, in the following a pattern-based symbolic algorithm for use the with computer-algebra-systems is brieﬂy explained [16], which allows to exactly evaluate a given second-order diﬀerential recursion in K-equations dg ðtÞ g ðtÞ¼ R g ðtÞ; g ðtÞ; ; n 0; ð22Þ nþ2 nþ1 n dt g ðtÞ¼ aðtÞ; ð23Þ g ðtÞ¼ bðtÞð24Þ up to a high number of coeﬃcients h (t) 2 R , n 1. The idea is based on sequential evaluation of the recursion combined with rule- and pattern-based routines to simplify the results. The respective computational procedure is schematically summarized below. Algorithm 1: For the pattern-based sequential evaluation of the second-order diﬀerential recursion (22)–(24) MATHEMATICA notation is utilized, i.e. ‘(*. . .*)’ denotes a comment, ‘a ! b’ a rule, and ‘/.’ a replacement or substitution operation: 226 T. Meurer and M. Zeitz Require : n > 0 max ð Initialize list Þ repList ¼fg ð Initialize rules Þ j j j d aðtÞ d bðtÞ d g repRule ¼ ! g ; ! g ; ! g 0;j 1;j i; j j j j dt dt dt i2N;j2N for n ¼ 0to n do max n n mod 2 for m ¼ 0to n do max ð Differentiate and, substitute Þ if m ¼ 0 then g ¼ g n;m n else dg n;m1 g ¼ n;m dt end if ð Apply rules Þ g ¼ g =:repRule=:repList n;m n;m ð Append result Þ append g to repList n;m end for end for The algorithm is initialized by (22)–(24) and the user-deﬁned speciﬁcation of the desired number of coeﬃcients n 2 N. This directly allows to determine the number of max derivatives of each coeﬃcient necessary to compute the coeﬃcients h (t) for n ¼ 0,1,..., n . The corresponding computations are performed within the inner for-loop. For this, max the list repList of derivatives h , m ¼ 0, 1,..., n –(n – n mod 2)/2 of the n-th n,m max m m coeﬃcient h is built-up, with any appearing derivative of the coeﬃcients d h (t)/dt with n j j 5 n being substituted by the state h . The respective rule is applied using repRule.Asa j,m result of the evaluation of the outer for-loop, a table of equations for the states h , n ¼ 0, n,m 1,..., n , m ¼ 0, 1, . . . , n –(n – n mod 2)/2 is obtained. max max The great advantage of this approach is related to the introduction of intermediate states, namely Z with m 4 0, such that the resulting equations can be evaluated in a n,m ‘top-down’ procedure with rather simple expressions for each state. In particular, since only symbolic computations are performed using computer-algebra-systems, the determined solution is exact. An eﬃcient numerical evaluation is possible by converting the MATHEMATICA results, e.g. to MATLAB C-Mex or standard C/Cþþ codes. 5. Feedforward tracking control using summability techniques for the tubular reactor model To illustrate the applicability of the approach combining formal power series and (N,x)- approximate k-summation, recall the tubular reactor model (2)–(4). As brieﬂy veriﬁed in Mathematical and Computer Modelling of Dynamical Systems 227 Figure 4. Comparison of numerical results for open-loop boundary control of the tubular reactor (2)–(4) using partial summation (pSum) and (N, x)-approximate k-summation (kSum) with N ¼ 40 series coeﬃcients. Left: input trajectory u*(t); right: output trajectory y(t) compared with desired y*(t). Model parameters: p ¼ 71.25, p ¼ 1.1, p ¼ 70.42 with Pe and summation parameters for 1 2 3 (21) as indicated. The L -error e ¼ky*(t)– y(t)k between desired and obtained output is provided 2 2 for comparison purposes. Note that for partial summation, numerical results when applying the respective feedforward control u*(t) to the tubular reactor model cannot be determined. 228 T. Meurer and M. Zeitz Section 2.2, uniform convergence of the series parametrization relates system parameters and trajectory planning in a rather restrictive manner. Numerical results when applying the input u*(t) determined from (8) by both partial summation (pSum) and (N,x)-approximate k-summation (kSum) for diﬀerent values of the Peclet-number Pe to a ‘method-of-lines’ discretized simulation model of (2)–(4) are shown in Figure 4. For this, the parameters of the desired trajectory (15) for the output (15) are 0 T assigned as y ¼ 0:5; y ¼ 0:05, and s ¼ 3/4 with a transition time of T ¼ 4. Note that for S S the present choice of parameters p , j ¼ 1, 2, 3 (corresponding to b (B) ¼ 71.0, j 0 b (B) ¼ 0.75 in (1)) the respective domain of convergence is depicted in Figure 2(b). However, since at ﬁrst the Pe-numbers are chosen outside the depicted domain and since secondly the chosen Gevrey order of y*(t) evaluates to a ¼ 1 þ 1/s ¼ 7/3 4 2, uniform convergence of the parametrized series (8) cannot be ensured. This is conﬁrmed by the simulation results in Figure 4 when applying partial summation, where obviously no meaningful solution can be obtained when considering moderate and large Peclet-numbers. The results can be greatly enhanced by applying the (N,x)-approximate k-summation (21). Obviously, accurate tracking in open-loop is achieved in the scenarios with Pe- number increasing from Pe ¼ 5 in Figure 4(a) to Pe ¼ 1000 in Figure 4(d). The tracking performance is thereby evaluated in terms of the L -error e ¼ky*(t)– y(t)k between 2 2 desired and obtained output. Obviously for all considered scenarios, highly accurate trajectory tracking can be achieved. In addition to the illustration of the tracking performance, Figure 5 shows the calculated concentration proﬁle x(z, t) over the (z, t)- domain for the open-loop transition between the respective initial and ﬁnal stationary 0 T proﬁles x ðzÞ and x ðzÞ for the scenario of Figure 4(c) with Pe ¼ 10. Clearly, the desired S S transition is achieved in the prescribed time-interval t 2 [0,T]. For large Pe-number, the dynamics of the DPS (2)–(4) is dominated by the convection and the reaction term and hence in the limit as Pe ! ? the DPS tends to a non-linear delay system. In this case, due to the resulting wave dynamics, it can be easily veriﬁed, e.g. by the method of characteristics, that there exists a minimal control time of t ¼ 1 which min has to be taken into account for trajectory planning. As can be seen from Figure 4(d), the resulting shift between input u*(t) and output y*(t), i.e. the control action has to start advanced by t , is approximately realized by applying the (N,x)-approximate min k-summation with suitably chosen parameters x and k. Figure 5. Concentration proﬁle x(z,t) over the (z,t)-domain for the open-loop transition between 0 T stationary proﬁles x ðzÞ! x ðzÞ within the time-interval t 2 [0,T], T ¼ 4. Model and summation S S parameters correspond to those of Figure 4 (b) for Pe ¼ 10. Mathematical and Computer Modelling of Dynamical Systems 229 6. Conclusions As is illustrated for the simpliﬁed distributed-parameter model of a tubular reactor, the formal power series provides a systematic approach for model inversion and trajectory planning for boundary controlled parabolic DPS. For this, a major enhancement of the range of applicability of the design approach is obtained by combining the formal series parametrization of state and input in terms of a basic output with appropriate summability methods. This in particular allows to use slowly converging or possibly diverging series for the trajectory planning and the solution of tracking control problems for DPS. Within the considered framework, more involved boundary controlled DPS can be treated similarly as illustrated in [16,21]. It is shown, that the introduced (N,x)-approximate k-sum yields the prototype summation technique when studying tracking control problems. In view of applications, the determined inversion-based feedforward tracking control has to be supplemented by a feedback part in order to deal with model errors, exogenous disturbances, or unstable plant models. This leads to the so-called ‘two-degrees-of- freedom’ control design, where the feedforward and feedback control are designed separately. Thereby, the desired tracking behaviour is realized by the feed-forward part while the feedback part is designed to drive the tracking error asymptotically to zero. It is interesting to notice that, based on a re-interpretation of the power series solution, the considered approach also allows to determine a ﬁnite-dimensional design model which is suitable for the design of a ﬂatness-based feedback tracking controller and of an observer for proﬁle estimation for parabolic DPS [16,21]. Acknowledgement The authors gratefully acknowledge ﬁnancial support by the ‘‘Deutsche Forschungsgemeinschaft’’ (DFG) in the project ZE 163/7. Furthermore, the authors thank the anonymous reviewer for useful comments in order to improve the paper as well as for referring to the results summarized in Remark 1. References [1] K. Jensen and W. Ray, The bifurcation behavior of tubular reactors, Chem. Eng. Sci. 37 (1982), pp. 199–222. [2] J. Winkin, D. Dochain, and P. Ligarius, Dynamical analysis of distributed parameter tubular reactors, Automatica 36 (2000), pp. 349–361. [3] R. Curtain and H. Zwart, An Introduction to Inﬁnite-Dimensional Linear Systems Theory, Texts in Applied Mathematics 21, Springer-Verlag, New York, 1995. [4] A. Butkovskiy, Distributed Control Systems, Elsevier, New York, 1969. [5] J. Lions, Optimal Control of Systems Governed by Partial Diﬀerential Equations, Springer- Verlag, 1971. [6] W. Ray, Advanced Process Control, McGraw-Hill, New York, 1981. [7] A. 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Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis

**Published: ** Jun 1, 2008

**Keywords: **model inversion; distributed-parameter system; differential flatness; formal power series; summability methods; feedforward tracking control; trajectory planning; tubular reactor

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