A relaxation of the strangeness index

Transcription

1 echnical report from Automatic Control at Linköpings universitet A relaxation of the strangeness index Henrik idefelt, orkel Glad Division of Automatic Control tidefelt@isy.liu.se, torkel@isy.liu.se 22nd February 2010 Report no.: LiH-ISY-R-2932 Address: Department of Electrical Engineering Linköpings universitet SE Linköping, Sweden WWW: AUOMAIC CONROL REGLEREKNIK LINKÖPINGS UNIVERSIE echnical reports from the Automatic Control group in Linköping are available from

2 Abstract A new index closely related to the strangeness index of a differential-algebraic equation is presented. Basic properties of the strangeness index are shown to be valid also for the new index. he definition of the new index is conceptually simpler than that of the strangeness index, hence making it potentially better suited for both practical applications and theoretical developments. Keywords: Differential-algebraic equations, strangeness index

3 1 Introduction Kunkel and Mehrmann have developed a theory for analysis and numerical solution of differential-algebraic equations. he theory centers around the strangeness index, which differs from the differentiation index in that it does not consider the derivatives of the solution to be independent of the solution itself at each time instant. Instead, it takes the tangent space of the manifold of solutions into account, thereby reducing the number of dimensions in which the derivative has to be determined. he book Kunkel and Mehrmann (2006) covers the theory well and will be the predominant reference used in the paper. he numerical solution procedure applies to general nonlinear differential-algebraic equations of higher indices, and is currently the only one we know of that can handle such problems, let be that it does not provide a sensitivity analysis. Our interest in this matter is mostly due to this capability. he present paper presents highlights from the corresponding chapter in the first author s thesis idefelt (2009). 2 wo definitions In this section, two index definitions will be presented along with some basic properties of each. he one to be presented first is the strangeness index, found in Kunkel and Mehrmann (2006). he second, which is proposed as an alternative, is called the simplified strangeness index. Both are based on the derivative array equations. 2.1 Derivative array equations and the strangeness index As always when working with dae, it is crucial to be aware that the solutions are restricted to a manifold. In practice, one is interested in obtaining equations describing that manifold, and the way this is done in the present paper is by using the derivative array introduced in Campbell (1993). Consider the dae f ( x(t), x (t), t )! = 0 (1) Assuming sufficient differentiability of f and of x, the idea is that the original dae is completed with derivatives of the equations with respect to time. his will introduce higher order derivatives of the solution, but the key idea is that, given values of x(t), it suffices to be able to determine x (t) in order to compute a numerical solution to the equations. hat is, higher order derivatives such as x (t) may appear in the equations, but are not necessary to determine. If the completion procedure is continued until the derivative array equations are one-full with respect to x (t), the procedure has revealed the differentiation index of the dae. he meaning of one-full is defined in terms of the equation considered pointwise in time, so that a variable and its derivative become independent variables. We emphasize the independence by using the variable ẋ(t) instead of x (t), where the dot is just an ornament, while the prime is an operator. he equations are then said to be one-full if they determine ẋ(t) uniquely within some open ball, given x(t) and t. An equivalent characterization can be made in terms of the Jacobian of the derivative array with respect to its differentiated variables; then the equations are one-full if and only if row operations can bring the Jacobian into block diagonal form, with a non-singular block in the block column corresponding to derivatives with respect to ẋ(t) (clearly, this shows that it is possible to solve for ẋ(t) without knowing the variables corresponding to higher order derivatives of x at time t). However, instead of requiring that the completed equations be one-full, it turns out that there are good reasons for using the weaker requirement that the equations display the strangeness index instead. he definition of strangeness index will soon be considered in detail. It turns out that equations displaying the strangeness index determine x (t) uniquely if one takes into account the connection between x(t) and x (t) being imposed by the non-differential constraints which locally describe the solution manifold. Strangeness-free equations (strangeness index 0) are suitable for numerical integration.(kunkel and Mehrmann, 1996) In the sequel, it will be convenient to speak of properties which hold on non-empty open balls inside the set { ( L ) } νs = t, x, ẋ,..., ẋ ν S +1 : F νs ( x, ẋ,..., ẋ (νs+1), t ) =! 0 (2)

4 2.1 Definition (Strangeness index). he strangeness index ν S at ( t 0, x 0 ) is defined as the smallest number (or if no such number exists) such that the derivative array equations 1 satisfy the following properties on for some δ > 0. P1a [2.1 F νs ( t, x(t), x (t), x (2) (t),..., x (ν S+1) (t) )! = 0 (3) L νs { ( t, x, ẋ,..., ẋ ν S +1 ) : t B t0 (δ) x B x0 (δ) } } {{ } here shall exist a constant number n a such that the rank of M νs ( t, x, ẋ,..., ẋ (ν S+1) ) = b δ [ FνS ( t, x, ẋ,..., ẋ (ν S +1) ) ẋ... F νs ( t, x, ẋ,..., ẋ (ν S +1) ) ẋ (ν S +1) is pointwise equal to ( ν S +1 ) n x n a, and shall exist a smooth matrix-valued function Z 2 with n a pointwise linearly independent columns such that Z 2 M ν S = 0. P1b [2.1 Let n d = n x n a, and let A νs = Z 2 N ν S where N νs ( t, x, ẋ,..., ẋ (νs+1) ) = F ν S ( t x, ẋ,..., ẋ (νs+1) ) x hen the rank of A νs shall equal n a, and there shall exist a smooth matrix-valued function X with n d pointwise linearly independent columns such that A νs X = 0. P1c [2.1 he rank of 2 f X shall be full, and there shall exist a smooth matrix-valued function Z 1 with n d pointwise linearly independent columns such that Z1 2f X is non-singular. Assume that the strangeness index is finite. hen the dimension of the solution manifold is n d = n x n a. P1b [2.1 then states that it is possible to construct a local coordinate map x(t) = φ 1 ( x d (t), t ) with coordinates x d, determined by the partial differential equation 1 φ 1 ( x d (t), t )! = X( x(t), t ) (4) where the columns of X are smooth functions and pointwise linearly independent. he columns of X can be selected as an orthonormal basis for the right null space of the matrix A, and the local coordinates x d are denoted the dynamic variables. he last property, P1c [2.1, is finally there to ensure that the time derivative of the local coordinates on the solution manifold are determined by the original equation (1). Replacing (1) by an equation with residual expressed only through the dynamic variables, f d ( x d, ẋ d, t ) = f ( φ 1 ( x d, t ), 1 φ 1 ( x d, t ) ẋ d + 2 φ 1 ( x d, t ), t ) (5) property P1c [2.1 states that the Jacobian with respect to ẋ d, 2 f d ( x d, ẋ d, t ) = 2 f ( φ 1 ( x d, t ), 1 φ 1 ( x d, t ) ẋ d + 2 φ 1 ( x d, t ), t ) X (6) is full-rank. Since there are only n d derivatives to be determined, and there are n x equations, there are n a more equations than unknowns. he property P1c [2.1 also states that n d linear combinations, given by the columns of Z 1, of the equations in (1) can be chosen smoothly and linearly independent (and hence orthonormal), so that these linear combinations are sufficient to determine the time derivatives of the dynamic variables. We now end this section with a lemma that will be useful later. 2.2 Lemma. If it is known that ν S ˆν, and the matrix [ N ˆν M ˆν does not have full row rank, then νs =. Proof: Let i ˆν. he upper part of [ N i M i equals [ N ˆν M ˆν 0, so [ N i M i cannot have full row rank. It follows that P1b [2.1 cannot be satisfied for ν S = i. 1 he notation is defined such that x (1) = x. 2

5 2.2 he simplified strangeness index By discretizing the derivatives (using a bdf method) in the original equation (1) (and scaling the equations by the step length), we get that the gradient of these equations with respect to x tends to 2 f ( x, ẋ, t ) as the step length tends to zero. Hence, joining these equations with the full derivative array equations (where no derivatives are discretized) yields a set of equations which (locally) shall determine x uniquely. his leads to the following definition. 2.3 Definition (Simplified strangeness-index). he simplified strangeness index ν q at ( t 0, x 0 ) is defined as the smallest number (or if no such number exists) such that the derivative array equations satisfy the following property on F νq ( t, x(t), x (t), x (2) (t),..., x (ν q+1) (t) )! = 0 (7) L νq { ( t, x, ẋ,..., ẋ ν q +1 ) : t B t0 (δ) x B x0 (δ) } } {{ } for some δ > 0. P2 [2.3 Let H νq = f ( x, ẋ, t ) ẋ F ν q ( t, x, ẋ,..., ẋ(ν q+1) ) 2 f 0 = N νq x M νq b δ F ν q ( t, x, ẋ,..., ẋ(ν q+1) ) ẋ... F ν q ( t, x, ẋ,..., ẋ(ν q+1) ) ẋ (ν q+1) where N νq and M νq are defined as in definition 2.1. hen it shall hold that I 0 [! 2 f 0 rank 2 f 0 = rank N νq M νq N νq M νq hat is, the basis vectors corresponding to x shall be in the span of the rows of H νq, which may be recognized as the property of H νq being one-full. he property P2 [2.3 can be interpreted as that there is no freedom in the x components of the solution to ( h f ( x, 1 h ( x ) q 1 x ), t )! F νq ( t, x, ẋ,..., ẋ (νq+1) = 0 ) since adding additional equations for the x variables alone does not decrease the solution space of the linearized equations. For theoretic considerations, however, the continuous-time interpretation provided by lemma 4.4 below is more relevant. Of course, we must show what the simplified strangeness index is for the inevitable pendulum. Example 1 Let us consider the following familiar model of a pendulum. ξ ξ u u f y, ẏ, t = v v λ λ λ ξ u λ y g v ξ 2 + y 2 1 ξ u ẏ v We consider initial conditions where the pendulum is in motion and neither x nor y is zero. o check P2 [2.3 we look at the projection of a basis for the right null space of H i onto the space spanned by the basis vectors corresponding to x, for i = 1, 2,..., ν q. (he projection is implemented by just keeping the 3

6 five first entries of the vectors.) he basis for the null space is computed using Mathematica, and for i = 0, 1, 2 the projected basis vectors are, in order, , 0 0, 0, 0 0, 0, ξ y y ẏ ξ ξ y ẏ ξ ξ y Assuming that the symbolic null space computations are valid in some neighborhood of the initial conditions inside L i, it is seen that the λ component is undetermined for i = 0 and i = 1, and as all components are determined for i = 2 we get ν q = 2. 3 Relations In this section, the two indices ν S and ν q will be shown to be closely related. his is done by means of a matrix decomposition developed for this purpose. We first show the matrix decomposition, and then interpret the two definitions in terms of this decomposition. 3.1 Lemma. he matrix [ N M where N R k l, M R k k, rank M = k a, a 1, can be decomposed as Q [ [ [ 3, Σ 0 Q N M 3,2 0 = Q1,1 Q 1,2 A Σ 1 Q1,1 N Q 2,1 0 Q 2,2 In this decomposition, the left matrix is unitary, as are the diagonal blocks of the right matrix. he matrix Σ is a diagonal matrix of the non-zero singular values of M. he matrix A is square. 3.2 heorem. Definition 2.1 and definition 2.3 satisfy the relation ν S ν q. Proof: Suppose that the strangeness index is ν S and finite, as the infinite case is trivial. Let the matrices N and M in lemma 3.1 correspond to M νs and N νs as in definition 2.1. First, let us consider ν S in view of this decomposition. he left null space of M is spanned by Q 1,2, and making these linear combinations of N results in Q1,2 N = [ A Q3,1 Q3,2 Σ 1 Q1,1 N = [ A 0 [ Q3,1 Q 3,2 0 where A has full rank due to P1b [2.1. his matrix determines the tangent space of the non-differential constraints as being its null space, spanned by the independent columns of Q 3,2. Hence, we can parameterize x as x = Q 3,2 x d. urning to ν q, we follow the constructive interpretation of P2 [2.3 in section 2.2. he right null space of [ N M is spanned by the second and fourth rows of the right factor in the decomposition; [ ( ) x! N M = 0 y ( ) [ ( ) (8) x Q3,2 0 z1 z 1, z 2 : = y 0 Q 2,2 z 2 Extracting the part of this equation which only involves x, we find that it can be parameterized in z 1 alone, and since the columns of Q 3,2 are independent, we can use z 1 as dynamic variables; x = Q 3,2 x d. 4

7 Since the strangeness index is ν S, 2 f Q 3,2 has full column rank according to P1c [2.1. Hence, [ ( ) 2 f 0 x! = 0 N M y ( ) ( ) (9) x 0 z 2 : = y Q 2,2 z 2 which is exactly the condition captured by P2 [2.3. Since ν q is the smallest index such that this condition is satisfied, it is no greater than ν S. 3.3 heorem. Definition 2.1 and definition 2.3 satisfy the relation ν S { ν q, }, with ν S = ν q if and only if ν q = or the following property holds. P1 [3.3 he matrix [ N νq M νq has full row rank on the set Lνq b δ in definition 2.3. hat is, rank [ N νq M νq = ( νq + 1 ) n x (10) Proof: In view of theorem 3.2, the statement is trivial in the case ν q =. Hence, consider ν q <, in which case it shall to be shown that ν S = ν q when P1 [3.3 holds, and ν S = otherwise. he latter case follows immediately from lemma 2.2, so it remains to consider the case when P1 [3.3 holds. Due to theorem 3.2 it suffices to show ν S ν q. Let the matrices N and M in lemma 3.1 correspond to M νq and N νq as in definition 2.3. he rank condition (10) implies that A in lemma 3.1 is non-singular. Consider (8) and (9). Since adding the equation 2 f x =! 0 is sufficient to conclude x = 0 given x = Q 3,2 z 1, it is! seen that 2 f Q 3,2 z 1 = 0 must imply z1 = 0. his is only true if 2 f Q 3,2 has full column rank, which shows that P1c [2.1 holds. Since ν S is the smallest index such that this condition is satisfied, it is no greater than ν q. 4 Uniqueness and existence of solutions he present section gives a result corresponding to what Kunkel and Mehrmann (2006, theorem 4.13) states for the strangeness index. As the difference between the two index definitions is basically a matter of whether P1 [3.3 is required or not, the main ideas in Kunkel and Mehrmann (2006) apply here as well. 4.1 Lemma. If the simplified strangeness index ν q is finite, there exist matrix functions Z 1, Z 2, X, similar to those in definition 2.1. hey are all smooth with pointwise linearly independent columns, satisfying Z 2 M ν q = 0 and columns of Z 2 span left null space of M νq (11a) Z 2 N ν q X = 0 and columns of X span right null space of Z 2 N ν q (11b) Z 1 2f X is non-singular (11c) Proof: Using the decomposition of lemma 3.1, we may take Z 2 Q 1,2 and X = Q 3,2. As in the proof of theorem 3.3, (8) and (9) then imply that 2 f X has full column rank, and the existence of Z 1 follows. Multiplying the relations in (11) by smooth pointwise non-singular matrix functions shows that the matrix functions Z 1, Z 2, X are not unique, but they can be replaced by any smooth matrices with columns spanning the same linear spaces. For numerical purposes, the smooth Gram-Schmidt orthonormalization procedure may be used to obtain matrices with good numerical properties, while the theoretical argument of the present section benefits from another choice, to be derived next. Select the non-singular constant matrix P = [ P d P a such that Z 2 N νq P a is non-singular in a neighborhood of the initial conditions, and make a change of the un-dotted variables in L νq according to x = [ ( ) x P d P d a (12) x a 5

8 he following notation will turn out to be convenient later (note that N a ν q is non-singular) N d ν q = Z 2 N νq P d N a ν q = Z 2 N νq P a (13) he next result corresponds to Kunkel and Mehrmann (2006, corollary 4.10) for the strangeness index. 4.2 Lemma. here exists a smooth function R such that inside L νq, in a neighborhood of the initial conditions. Proof: In L νq it holds that F νq = 0 and Z 2 M ν q = 0, and it follows that Z2 F ν q ẋ (1+) = Z2 x a = R( x d, t ) (14) F νq ẋ (1+) + Z 2 ẋ (1+) F ν q = 0 Hence, the construction of Z 2 is such that Z 2 F ν q only depends on t and x, and the change of variables (12) was selected so that the part of the Jacobian corresponding to x a is non-singular. It follows that x a can be expressed locally as a function of x d and t. We now introduce the function φ 1 to describe the local parameterization of x using the coordinates x d and t, ( ) x = φ 1 ( x d, t ) = x P d (15) R( x d, t ) and the next lemma shows an important coupling between φ 1 and lemma Lemma. he matrix X in lemma 4.1 can be chosen in the form [ I ˆX = P = 1 R( x d, t ) 1 φ 1 ( x d, t ) (16) Proof: Clearly, the columns are linearly independent and smooth. By verifying that the matrix is in the right null space of Z 2 N ν q we will show that its column spans the same linear space as X. It will then follow that X and ˆX are related by a relation in the form ˆX = X W for some smooth non-singular matrix function W. Using the form X W then shows that (11c) is also satisfied. Hence, it remains to show that ˆX is in the right null space of Z 2 N ν q. Using (14) and allowing also the dotted variables ẋ (1+) to depend on x d in (suppressing arguments) it follows that Z2 F νq F x νq + Z2 + d x d F ν q Z 2 F ν q x d = 0 Z2 F νq F x νq + Z2 a x x a + a x d Here, F νq = 0 and Z2 = Z ẋ (1+) 2 M ν q = 0 implies that Z 2 F νq x d + Z 2 F νq x a 1 R = Z 2 2F νq [ Pd P a Z2 ẋ (1+) F ν q + Z2 F νq ẋ (1+) [ I = Z 1 R 2 N ν q ˆX =! 0 ẋ(1+) x d! = 0 Back in section 2.2 it was indicated that we would be able to show that a finite simplified strangeness index implies local uniqueness of solutions. With lemma 3.1 at our disposal this statement can now be shown rather easily. 4.4 Lemma. If the simplified strangeness index is finite and x is a solution to the dae for some initial conditions in L νq b δ, then the solution x is locally unique. 6

9 Proof: Using the parameterization of x given by (15), it suffices to show that the coordinates x d are uniquely defined. By the smoothness assumptions and the analytic implicit function theorem, Hörmander (1966), showing that x d (t) is uniquely determined given x d(t) and t will be sufficient, since then the corresponding ode will have a right hand side which is continuously differentiable, and hence locally Lipschitz on any compact set. One may then complete the argument by applying a basic local uniqueness theorem for ode, such as Coddington and Levinson (1985, theorem 2.2)). Reusing (5) for the current context, x d (t) is seen to be uniquely determined if 2f d ( x d, ẋ d, t ) is non-singular (in some neighborhood L νq b δ of the initial conditions). Identifying (6) in (11c), lemma 4.3 completes the proof. With ˆX according to (16) it follows that Z 2 N ν q ˆX = N d ν q + N a ν q 1 R! = 0 (17) using the notation (13). Before stating the main theorem of the section we derive one more equation. Using (14) and allowing also the dotted variables ẋ (1+) to depend on t in (suppressing arguments) it follows that Z 2 Z 2 F νq t! = 0 ( 1 F νq + 2 F νq 2 φ 1 ẋ (1+) ) + M νq = Z t 2( 1 F νq + 2 F νq 2 φ ) 1! = 0 (18) 4.5 heorem. Consider a sufficiently smooth dae (1), repeated here, f ( x(t), x (t), t )! = 0 (1) with finite simplified strangeness index ν q and where the un-dotted variables in L νq form a manifold of dimension n d. If the set where P2 [2.3 holds is the projection of a similar b δ+ L νq +1, and P2 [2.3 also holds on b δ+ L νq +1 with the same dimension n d, then there is a unique solution to (1) for any initial conditions in b δ+ L νq +1. Proof: Considering how F νq +1 is obtained form F νq, it is seen that the equality F νq +1 = 0 can be written 1 F νq + 2 F νq ẋ + 3+ F νq ẋ (2+) = 0 Multiplying by Z 2 from the left and identifying the expressions for N ν q and M νq, one obtains Z 2( 1 F νq + 2 F νq ẋ ) = 0 Using (18) and the change of variables (compare (12)) leads to (using the notation introduced in (13)) Using (15) and (17) yields ẋ = [ P d ( ) ẋ P d a ẋa [ N d νq N a ν q ( P 1 2 φ 1 + ) ) (ẋd = 0 ẋ a N a ν q 2 R( x d, t ) N a ν q 1 R( x d, t ) ẋ d + N a ν q ẋ a = 0 (19) and since Nν a q is non-singular, it must hold that ) ( ) (ẋd I ẋ = P = P ẋ ẋ a 1 R( x d, t ) d + P ( ) 0 2 R( x d, t ) = 1 φ 1 ( x d, t ) ẋ d + 2 φ 1 ( x d, t ) Since f ( x, ẋ, t )! = 0 holds by definition on L νq, it follows that f ( φ 1 ( x d, t ), 1 φ 1 ( x d, t ) ẋ d + 2 φ 1 ( x d, t ), t ) = 0 7

10 where ẋ d is uniquely determined given x d and t by (11c) with 1 φ 1 = ˆX in place of X. Hence, the dae f ( φ 1 ( x d (t), t ), 1 φ 1 ( x d (t), t ) x d (t) + 2φ 1 ( x d (t), t ), t ) = 0 has a (locally unique) solution and the trajectory generated by x(t) = φ 1 ( x d (t), t ) is a solution to the original dae (1). 5 Conclusions and future work In our view, a simpler way of computing the strangeness index has been proposed. While the original definition follows a three step procedure, the proposed definition has just one step. Once the index has been determined according to the new definition, it is known that the original definition leads to the same or an infinite index, and there is a simple test that distinguishes the two cases. he new index definition is also appealing due to its immediate interpretation from a numerical integration perspective. Analogues of central results for the original strangeness index have been derived for the simplified strangeness index. In particular, it has been shown that a finite simplified strangeness index implies that if a solution exists, it will be unique, and existence of a solution can be established by checking the property that defines the index for two successive values of the index parameter. An important aspect of the analysis of the strangeness index provided in Kunkel and Mehrmann (2006, chapter 4) is that the strangeness index is shown to be invariant under some transformations of the equations which are known to yield equivalent formulations of the same problem. It is an important topic for future research to find out whether the simplified strangeness index is also invariant under these transformations. Another interesting topic for future work is to seek examples where ν q ν S in order to get a better understanding of this exceptional case. 6 Acknowledgment he authors would like to acknowledge Ulf Jönsson at the Royal Institute of echnology, Sweden, for strengthening theorem 3.3. References Stephen L. Campbell. Least squares completions for nonlinear differential algebraic equations. Numerische Mathematik, 65(1):77 94, December Earl A. Coddington and Norman Levinson. heory of ordinary differential equations. Robert E. Krieger Publishing Company, Inc., third edition, Lars Hörmander. An introduction to complex analysis in several variables. he University Series in Higher Mathematics. D. Van Nostrand, Princeton, New Jersey, Peter Kunkel and Volker Mehrmann. A new class of discretization methods for the solution of linear differential-algebraic equations with variable coefficients. SIAM Journal on Numerical Analysis, 33(5): , October Peter Kunkel and Volker Mehrmann. Regular solutions of nonlinear differential-algebraic equations. Numerische Mathematik, 79(4): , June Peter Kunkel and Volker Mehrmann. Differential-algebraic equations, analysis and numerical solution. European Mathematical Society, Henrik idefelt. Differential-algebraic equations and matrix-valued singular perturbation. PhD thesis, Linköping University,

11 Avdelning, Institution Division, Department Datum Date Division of Automatic Control Department of Electrical Engineering Språk Language Rapporttyp Report category ISBN Svenska/Swedish Licentiatavhandling ISRN Engelska/English Examensarbete C-uppsats D-uppsats Övrig rapport Serietitel och serienummer itle of series, numbering ISSN URL för elektronisk version LiH-ISY-R-2932 itel itle A relaxation of the strangeness index Författare Author Henrik idefelt, orkel Glad Sammanfattning Abstract A new index closely related to the strangeness index of a differential-algebraic equation is presented. Basic properties of the strangeness index are shown to be valid also for the new index. he definition of the new index is conceptually simpler than that of the strangeness index, hence making it potentially better suited for both practical applications and theoretical developments. Nyckelord Keywords Differential-algebraic equations, strangeness index