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Mathematical and Computer Modelling of Dynamical Systems 1387-3954/01/0702-159$16.00 2001, Vol. 7, No. 2, pp. 159±172 Swets & Zeitlinger K. SCHLACHER and A. KUGI ABSTRACT Many problems in mathematical modeling of lumped parameter systems lead to sets of mixed ordinary differential and algebraic equations. A natural generalization are so called descriptor systems or sets of implicit ordinary differential equations, which are linear in the derivatives. This contribution deals with variational problems for descriptor systems. Using the mathema- tical language of Pfaf®an systems, we derive a canonical form of a descriptor system, which can be converted to an explicit control system in principle. Since the proposed approach does not use this transform explicitly, the Euler Lagrange and Hamilton Jacobi equations of the varia- tional problem are derivable by pure algebraic manipulations. In addition, this approach leads to computer algebra based algorithms, which are needed to perform the required calculations ef®ciently. 1 INTRODUCTION In the mathematical modeling of lumped parameter systems it has turned out that the DAE-system or descriptor system approach  is a very natural one. A DAE-system is a set of n ordinary differential equations combined with m pure algebraic relations of the form w _ f t; w; v ; 0 g t; w; v 1 with 1; .. . ; n, 1; .. . ; m. A more general class of systems, the class of descriptor systems is formed by the set of equations i i e w w _ f t; w; v 2 with a singular matrixe . Although both types of constraint dynamic systems show some similarity to an explicit control system x f t; x; u 3 Department of Automatic Control, Johannes Kepler University of Linz, Altenbergerstraûe 69, A-4040 Linz, Austria. E-mail: schlacher[kugi]@mechatronik.uni-linz.ac.at 160 K. SCHLACHER AND A. KUGI n m with the state x 2 X R and the input u 2 U R , it is straightforward to see that neither w is the usual state of a dynamic system nor u the input of a dynamic system in general. In [6, 7] it was shown that the investigations of (1) and (2) become easier, if we give up to distinguish between w and v and investigate systems of the type e i n z z _ m z; 1; .. . ; n ; i 1; .. . ; n ; 4 e e z where w, v are combined to the new variable z w; v . Like we did in (2, 4), we will use Einstein's abbreviations for sums, throughout this contribution and suppress the range of an index, whenever it is clear from the context. For the sake of simplicity we assume also that all functions are smooth. The main goal of the contribution is the geometric approach for the determination of the variational equations for (4) with an objective functional lt ; z dt 5 in form of the Euler Lagrange and the Hamilton Jacobi equations. As a special case, we will present the results for a linear and time invariant system of the type (4) and a standard quadratic functional. This paper is organized as follows. In the second section, we introduce a canonical form for descriptor systems and present an algorithm, which allows us to transform the descriptor system (4) to this form under some mild rank conditions. In the third section, we treat the variational problem (4) and show that Pfaf®an systems are a useful tool to derive the Euler Lagrange and the Hamilton Jacobi equations ef®ciently. Finally, this contribution ®nishes with some conclusions. 2 A CANONICAL FORM Before starting our investigations of equation (4) we introduce some useful notations. t 2 T R is the independent variable and z 2 Z R denotes the dependent variables. Their total manifold is given by E T Z, where we use the coordinates t; z locally. This choice is admissible, since all our results will be local ones except the results for the linear and time invariant scenario. DESCRIPTOR-SYSTEMS AND OPTIMAL CONTROL 161 A useful tool for our investigations are jet coordinates. The ®rst jet bundle ofE with coordinates t; z; z _ is speci®ed by JE. It is worth mentioning that for the present z _ is nothing else than the name of a coordinate. A ®rst observation is that (4) is an equation in the coordinates of JE. From now on, we assume that (4) de®nes a submanifold S of JE, at least locally. Of course, this submanifold is somehow related to a set of differential equations. Let denote a section t; z t of E, then its prolongation to a section of JE is denoted 1 i i by t; z; z _ t with z _ @ . Therefore, a solution of (4) is nothing 1 else than a section of E, whose prolongation meets the relations n @ m . At the ®rst glance, this notation looks arti®cial, its main advantage is the clear separation between the algebraic structure of (4) and the solutions of (4). To express the geometric structure of (4) ef®ciently, we need also some standard notation from differential geometry (e.g., see [2, 3]). Let M be a smooth manifold, then TM denotes the tangent bundle of M, ^ M the exterior k bundle and^ M the exterior algebra overM. d is the exterior derivative d : ^ M !^ M and i : T M ^ M ! ^ M is k k1 k1 k the interior product written as i ! , X2T M , !2^ M . As usual, we k1 identify ^ M with M and ^ M with the cotangent bundle T M of 0 1 M. The Lie derivative of !2^ M along the ®eld f 2T M is written as f ! . Let us take a look to the explicit control system (3) with the vector-®eld v f @ @ x t ofTE with E T X U. Obviously, the relations i du 0 are met for k 1; .. . ; m. Equivalently, (3) describes a submanifold of T T X parametrized by u. We gain another view, if we look at the submanifold S of JE generated by (3). Obviously, S is also a submanifold of JT X U, since (3) is independent of u _ . Furthermore, let E denote the tangent-space of 1 n S at a point p, then the n-form d x _ ^^ d x _ restricted to E does not vanish for any p. Obviously, a solution of (3) depends also on some constants, the initial values of the state, and some free functions, the input. Both properties, presented just now, allow us to determine their numbers. Roughly speaking, we will try to discover similar properties of (4) in the following subsections. Throughout this contribution, we assume that all functions being involved are suf®ciently enough continuously differentiable to avoid mathematical subtleties. 162 K. SCHLACHER AND A. KUGI 2.1 Linear Systems Let us consider the linear and time invariant system e i e i n z _ m z m z; 1; .. . ; n ; i 1; .. . ; n : 6 e e z i i AgainE T Z denotes the total manifold and JE its ®rst jet bundle. It is e e worth mentioning that the real valued matrices n , m have no intrinsic i i meaning. Only two observations are critical for the following. The ®rst one is that the system (6) contains at least one additional constraint m 0, if there exists a non-trivial solution of the equations n 0; i 1; .. . ; n : 7 Introducing the total time derivative, the special vector ®eld d z _ @ @ t i t with @ @ i , we see immediately that the equation i z i i e e d m z m z _ 0 8 e i e i is a consequence of the constraint (7). We call such a system ill-posed, because it contains algebraic constraints for the dependent variables. A system of the type (6) is called well-posed, if no hidden algebraic constraints for the dependent variables exist. Let us assume that (6) is well-posed and has the special form i i x x x d ' n z _ m z ; 1; .. . ; n t x x i i d ' 0; n 1; .. . ; n ; 9 t s x e where the equations d ' 0 follow from the pure linear algebraic s i constraints ' ' z 0. The second observation is that the transform ', x; s; u ' z , ÿn x x s x s x ' ; s ' ÿn u e u u '; n 1; .. . ; n 10 u e z with n n n allows us to rewrite (9) in the explicit form e x s x x x _ m z ÿ1 z' x;s;u ÿn s x s _ 0 11 ÿn s x with the simple constraints s 0. Of course, the linear functions i n u x ' ' z must be chosen such that (10) is an isomorphism. Now, x 2 R is i DESCRIPTOR-SYSTEMS AND OPTIMAL CONTROL 163 n ÿn z e the state and u 2 R the input of an explicit control system, because the n ÿn e x constraint s 2 R can be eliminated in a trivial way. Since we can use the same algorithm to transform an ill-posed nonlinear system of the type (4) to a well posed one, we will postpone this problem to the next section. The results above need some remarks. Although the matrix n of (6) has no intrinsic meaning, this is not true any more for n of (9). The map ' : Z ! X can be considered as a section of E Z X. We use the coordinates z; x; x , I fg 1; .. . ; n for the ®rst jet space of E, denoted by I z 1 x x x JE. The prolongation ' of ' to JZ X is given by x @ ' . x x Obviously, we have @ ' n , and the matrix n describes a map, which i i comes from a prolongation of a section of the total manifold. Obviously, the s u same arguments are valid for ' , ' , too. The interesting fact is that these constructions work also in the nonlinear case, if it is adapted in the right way. The well known obstacle in the nonlinear case is the fact that the matrix n may fail to be the Jacobian of an ordinary section. 2.2 Descriptor Systems Like in the linear case, only two observations are critical for the following. The ®rst one, the counterpart to (7), (8), is that a non-trivial solution of the equations z n z 0; i 1; .. . ; n 12 e i generates an additional constraint m 0 of (4). Again the differential equation d m 0 13 is a consequence of (12). In contrast to the linear and time invariant case the rank of the matrix n may depend on z. Therefore, we restrict all our investigations to the generic case and assume that the rank of n is constant in the neighborhood of a generic point z. Again, we call such a system ill- posed, if it contains algebraic constraints for the dependent variables, and well-posed, if it does not. The second observation is based on the well posed system in the special form x i n z _ m ; 1; .. . ; n x x d ' 0; n 1; .. . ; n ; 14 t s x e where the equations d ' 0 follow from the pure algebraic constraints x x s x s ' 0. Let us assume that there exist functions ' , ' , a , ^ a , x x 164 K. SCHLACHER AND A. KUGI x x x x ^ a a with the Kronecker±symbol and ; 1; .. . ; n de®ned on x x x x x x x E such that x x x s x a d ' a d ' n z _ 15 t t x s is met with functionally independent functions ' . Then there exists an invertible map ', x; s; u ' z , ÿn x x s x s x ' ; s ' ÿn u e u u ' ; n 1; .. . ; n 16 u e z with n n n such that (14) together with (16) can be rewritten as e x s ÿ1 x x x _ ^ a m ' x; s; u ÿn s x s _ 0 17 ÿn s x with the simple constraints s 0. Obviously, there exists a signi®cant difference to the linear and time invariant case, since the integrability conditions (15) are necessary and locally suf®cient conditions for the existence of the map (16). This is just the obstacle mentioned in the subsection above. For the derivation of an algorithm, which transforms an ill-posed system to a well-posed one, it is advantageous to rewrite the system (4) as a Pfaf®an x i x s s system P fg n dz ÿ m dt; d' ;'fg , n dz ÿ m dt; 1; .. . ; n x x d' ; n 1; .. . ; n : 18 s x e x i A solution of (18) is a section t t; zt of E such that n dzÿ x s s m dt d' 0, ' t 0 is met. Here : ^ E !^ T 1 1 denotes the pull back by . Obviously, the solutions are constraint to the submanifoldNfg t; z 2Ej ' z 0 . Let us assume that the relation d' 6 0 19 as well as ^ ^ s x x d' ^ n dz ÿ m dt 6 0 20 are met in addition, then we call the system adjusted. Adjusting the system needs two steps. In the ®rst step, we eliminate a minimal number of functions DESCRIPTOR-SYSTEMS AND OPTIMAL CONTROL 165 ' such that (19) is met. In the second step, we eliminate a minimal number of further forms to meet (20). The following algorithm will perform the required transformation. 0 x i 1. Start with P fg n dz ÿ m dt ;fg . k k1 k 2. Adjust P to derive P and determine n , n for P . x e V V x i 3. If n dz^ d' 6 0 then stop. 4. Otherwise the system contains new constraints of the type (12). Add the constraints to the system and go to 2. This algorithm generates a sequence of Pfaf®an systems P , k 1; .. . ; l constrained to manifoldsN such that the number n is strictly decreasing and 0 1 l N N N is ful®lled. Obviously, the last system P is well-posed, since the condition for the termination guarantees that (12) cannot be met. Now some facts are worth mentioning. The condition (15) can be checked easily, if we consider the Pfaf®an system (18), because the Theorem of Frobenius (e.g. ) states that equivalently the relations i i x x s d n dz 0 mod n dz ; d' 21 i i must be met (). Furthermore, the 1-forms ! , x s s u u ! n dz;! d' ;! d' 22 together with dt form a basis B fg dt;! ofT E. The canonical dual basis B ofTE will be denoted from now on by B @ ;@ ;@ ;@ : 23 fg x s u It is easy to see that B can be determined without knowing the isomorphism (16) and that the distributions spanfg @ and spanfg @ are involutive. x u x i Although the forms n dz have no intrinsic meaning, this is not true any more, if (21) is met. Analogously to the linear and time invariant case, the relation x x x x s d' ^ a n dz ÿ a d' x s 1 shows that the matrix n is related to the section ' of JE, E Z X such that n is part of the Jacobian of an ordinary section. i 166 K. SCHLACHER AND A. KUGI The presented algorithm preserves the integrability conditions (21). Since systems like (1) ful®ll always these conditions, there is no need to check them after the application of the presented algorithm. Therefore, DAE-systems form an important subclass of the descriptor systems. An interesting problem is to ®nd a canonical form for a descriptor system (4) that does not meet the integrability conditions (15). If a dynamic extension of the system by adding more differential equations is admissible then a trivial solution is given by n z _ m ; 1; .. . ; n x x d ' 0; n 1; .. . ; n t s x e ÿn u u e d ' z ; n 1; .. . ; n t u e z with the extended set of dependent coordinates z; z . Obviously, this system ful®lls the integrability conditions, but unfortunately this is not the only solution to this problem. A challenging problem is to determine the minimal number of differential equations that must be added such that the extended system meets the integrability conditions. According to the knowledge of the authors, this is still an open problem. Last but not least, it should be mentioned that a computer algebra system is indispensable to execute the proposed algorithm . 3 OPTIMAL CONTROL Optimal control for descriptor systems like (4) with an objective function like (5) belongs to a class of very dif®cult problems and this general case will not be treated here. The situtation changes signi®cantly, if we restrict ourselves to well-posed systems (14) that meet the conditions (15) such that one can transform them to the explicit form (17) by (16). Of course, by application of this transform one converts the implicit problem to an explicit problem, which can be treated by standard tools. The disadvantage of this approach is that it is dif®cult to ®nd the transform (16), since it requires the solution of partial differential equations. We will show that one can obtain equivalent results without knowledge of (16), only the conditions (21) and (22) must be ful®lled. Let us consider the system (14) or equivalently the well-posed Pfaf®an system DESCRIPTOR-SYSTEMS AND OPTIMAL CONTROL 167 x i n dz ÿ m dt; 1; .. . ; n x x d' ; n 1; .. . ; n : 24 s x e 1 Let the section t; z of E be a solution of (14) or equivalenty meets x i x s n dz ÿ m dt d' 0. From (5) we derive the functional l t dt: 25 We look for the determining equations for a solution of (24), which minimizes (25). To complete the problem, we have to add suitable boundary conditions. For the sake of simplicity we consider ®xed terminal points t , t . 1 2 The standard approach to this variational problem starts with the set of all 1-dimensional point transforms ' :E!E, such that the independent coordinate t as well as the terminal points remain unchanged , . Roughly speaking, such a map ' transforms a solution of (24) into another one. In addition, the minimizing section must ful®ll the inequality ' . Now, it is well known that the determining equations for the extremal solution can be derived from the conditions given by 1 i 1 x s v n z _ ÿ m 0; v d ' 0; v dt 0 26 where v denotes all vector-®elds v2VE T E, i i i v V @ i; V @ ' ; 27 0 1 which are induced by the point transforms ', and v their prolongations to the 1st-jet space are given by 1 1 i i v ' V @ i d V @ i: 28 z z _ 0 We will not follow this approach, but we will study equivalently the well- posed Pfaf®an system (24). Now, one can show  that the equation (26) are equivalent to the set x x s ! l m dtÿ n dz ÿ d' x x i s vcd! j 0; 29 which depends only on the vector-®elds v of (27), but does not depend on their 1 prolongations v (see (28)) any more. The 1-form ! is de®ned on C 168 K. SCHLACHER AND A. KUGI ZE R with local coordinates t; z; . The notation !j means the restriction of the form ! to the submanifold t ; t Z . Still, there 1 2 remains the problem to ®nd the equations, which determine the optimal solution. It is worth mentioning that the variables do not reduce to the Lagrange multipliers necessarily. A basis of T E and one ofTE are given by (22) and by (23), respectively. Obviously, one can extend them to T Z and TZ and we get B fg dt;! x x s s u u ! n dz;! d' ;! d' x s ! d ;! d 30 ÿn ÿn ÿn z z x x s with n 1; .. . ; n n , n n 1; .. . ; n n for T Z. The z z x z x z e x s canonical dual basis ofTZ is denoted by no B @ ;@ ;@ ;@ ;@ ;@ : 31 x s u x s Now it is straightforward to see that B fg @ ;@ ;@ is a basis for the x s u variational ®eld v of (27), since the independent variable t remains unchanged. This fact will allow us to derive the Euler Lagrange equations of the variational problem. Since we are interested mainly in the derivation of the Euler Lagrange and Hamilton Jacobi equations, we will neither discuss the second order conditions nor the problem of conjugate points, since the main goal of this contribution is the algorithmic derivation of the determining equations for the optimal solution. Roughly speaking, we will not present results concerning the existence and uniqueness of solutions. 3.1 The Linear Case Let us consider the well-posed Pfaf®an system i i x x n dz ÿ m z dt; 1; .. . ; n x x i i s s d' ' dz; n 1; .. . ; n 32 s x e with ' 0 associated to (9) and the objective functional i j z q z dt 33 ij 0 DESCRIPTOR-SYSTEMS AND OPTIMAL CONTROL 169 with a positive semi-de®nite matrix q . Following (29), we get the forms ij i j i i x x s ! z q z m z dtÿ n dz ÿ d' C ij x x s i i ÿ ÿ j i i x x d! z q m dz m z d ^ dt C ij x i i x i i x s ÿ d ^ n dz ÿ d ^ ' dz 34 x i s i and derive the system ÿ ÿ j i i d! z q m i dz dt d @ C ij @ x i x x x ÿ ÿ j i i d! z q m i dz dt d @ C ij @ x s s i s ÿ ÿ j i i d! z q m i dz dt @ C ij @ x i u u i i x x i d! m z dtÿ n dz ; @ C i i i d! ÿ' dz 35 @ C of 1-forms, the Pfaf®an system of the Euler Lagrange equations. The associated descriptor system is given by i i x x n z _ m z i i ' z _ 0 ÿ ÿ j i ÿ z q m i dz ij @ x x i x ÿ ÿ j i ÿ z q m i dz ij @ s x i ÿ ÿ j i 0 z q m i dz ; 36 ij @ i u since we must have dt 6 0. This condition allows us to choose t as the independent variable. It is worth mentioning that the differential equations for the are explicit ones. This is a direct consequence of the choice for the basis (30) and (31). Let us consider a submanifold YZ such that j 6 0; V V j x i x s dt^ ! ^ ! and z q m i dz 0 are ful®lled. These ij @ x i x s x conditions allow us to choose t, x and s (see (10)) as independent coordinates to parametrizeY. If additionally d! j 0 holds, then there exists a function V on Y such that ! j dV 0 is met. Because of x s u dV @ V dt @ V! @ V! @ V! x s u 170 K. SCHLACHER AND A. KUGI and (34) we get i j i @ V z q z m z 0 t ij x i @ V; @ V;@ V 0: 37 x x s s u This set of equations together with the constraints ÿ ÿ j i i x s z q m i dz ;' z 0 38 ij @ x i i are nothing else than the Hamilton Jacobi equations of the variational problem. Now it is straightforward to look for a solution of the limit case t !1. Of course, it is easy to see that one can convert this problem to the standard LQR-problem by applying the transform (10). 3.2 The Descriptor Case Let us consider the well-posed Pfaf®an system (24) and the objective function (25). The equation (29) are given by x x s ! l m dtÿ n dz ÿ d' x x i s x x d! dl dm d m^ dt x x i i x x ÿ d ^ n dz ÿ d n dz ÿ d ^ d' 39 x i x i s in this case. Using the basis (30) and (31) for the variational ®eld, we derive the Pfaf®an system of the Euler Lagrange equations of this variational problem as x i i d! i dl dm dt d ÿ i d n dz @ C @ @ x x x i x x x x i i d! i dl dm dt d ÿ i d n dz @ C @ @ x s x i s s s x i i d! i dl dm dtÿ i d n dz @ C @ @ x x i u u u x i i d! m dtÿ n dz @ C i d! ÿd' : 40 @ C In general, (40) is a complicated descriptor system, but in contrast to the linear case, it is not in explicit form for the variables . Because of (21) we have x i x i x x s d n dz ^ n dz ^ d' ; i i x s DESCRIPTOR-SYSTEMS AND OPTIMAL CONTROL 171 and short calculations show that x i x x x i d n dz i m dtÿ @ @ > x x x x > x i x i x x x x s i d n dz i m dtÿ mod n dz ÿ m dt; d' @ @ i i s s x s x i ; x x i d n dz i m dt @ @ u u x is met. The combination of the equations above with (40) gives the ®nal set of the Euler Lagrange equations for this variational problem x i x x x m dtÿ n dz ; i dl $ dt d i x x x x x s x x d' ; i dl $ dt d x s x x x x x x i dl $ dt;$ dm ÿ m : 41 u x x i x x x s By construction we have 0 modfg n dz ÿ m dt; d' ; dt and x s x x therefore, there exist functions h , h such that x s x x h dt x i x x x s mod n dz ÿ m dt; d' : x x h dt s s This implies that we convert the forms of (41), which contain the quantities d to explicit differential equations in . The derivation of the Hamilton Jacobi equations is a straightforward task and follows directly from the linear case. Again we restrict the problem V V x s to the submanifold YZ with j 6 0, dt^ ! ^ ! , x s i dl $ 0, where d! j 0 is met. The required function V @ C x Y meets the equations @ V l m 0 @ V; @ V;@ V 0 42 x x s s u together with the constraints x s i dl $ ;' 0: 43 u i It is worth mentioning that the Euler Lagrange equations and the Hamilton Jacobi equations for a nonlinear H -or H -problem can be derived in a 2 1 straightforward manner, if the function l is replaced by a suitable objective function and additional information for the descriptor state w and descriptor input v (see (2)) is available. 172 K. SCHLACHER AND A. KUGI 4 CONCLUSIONS This contribution has shown that there is no essential difference in the calculus of variations for explicit control systems and descriptor systems, which are transformable to explicit systems in principle. Based on the presented geometric framework using the mathematical language of Pfaf®an systems, the variational equations in form of the Euler Lagrange or the Hamilton Jacobi equations can be determined in a straightforward manner in the neighborhood of generic points. This approach requires the calculation of a canonical Pfaf®an system associated to the descriptor system, which offers the identi®cation of the real input and the real state of a control system. Based on this form, we can derive the equations of a variational problem constrained by a descriptor system by pure algebraic manipulations only, which can be done by a computer algebra system. REFERENCES 1. Brenan, K.E., Campbell, S.L. and Petzold, L.R.: Numerical Solution of Initial-Value Problems in Differential Algebraic Equations. SIAM, 1996. 2. Frankel, Th.: The Geometry of Physics. Cambridge University Press, 1997. 3. Grif®ths, P.A.: Exterior Differential Systems and the Calculus of Variations. Birkha Èuser Verlag, 1983. 4. Haas, W., Schlacher, K. and Kugi, A.: A Software Package for the Analysis of DAE Control Systems. European Control Conference 99, Karlsruhe, 1999. 5. Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, 1993. 6. Schlacher, K., Kugi, A. and Haas, W.: Geometric Control of a Class of Nonlinear Descriptor Systems. NOLCOS, Enschede, Vol. 2, 1998, pp. 387±392. 7. Schlacher, K., Haas, W. and Kugi, A.: Ein Vorschlag fu È r eine Normalform von Deskrip- torsystemen, ZAMM, Angew. Math. Mech. 79, 1999, pp. S21±S24.
Mathematical and Computer Modelling of Dynamical Systems – Taylor & Francis
Published: Jun 1, 2001
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