2.2 Differential-Algebraic Equations
2.2.4 Remarks
with ˆxj−1 =xj0 and due to (2.34) we get E(t)( ˙ˆx + ˙ximp+X
i∈Z
(ˆxi(τi)−xˆi−1(τi))δτi) = A(t)(ˆx + ximp) +E(t) ˙xj0
−A(t)xj0+X
i6=j
bi+ bimp. Setting ˜x = x−xˆj−1 and ˜b = b−ˆbj this can be expressed in the form
E(t) ˙˜x =A(t)˜x + ˜b +E(t)xj,0δτj, x˜j−1 = 0, (2.36) wherexj,0 = ˆxj−1(τj). The initial condition does not occur as it is stated in the classical for-mulation, as we cannot prescribe values of distributions, but as part of the inhomogeneity.
The general formulation of an initial value problem
E(t) ˙x =A(t)x + b +E(t)xj,0δτj, xj−1 = 0, (2.37) suggests that for sufficiently smooth b the smoothness of x will depend on the initial condition. Thus, the impulsive behavior and the future smooth development of the system does not depend on the whole history but only on the initial condition.
Proof. See [82, Corollary 3.26].
Thus, the consequence of the constant rank assumption in Theorem 2.34 is that the strangeness index is defined on a dense subset of a given closed interval I ⊆ R. Note, that the proof of Theorem 2.50 and Corollary 2.51 is given in [82] only for the case of complex matrix-valued functions. Nevertheless, the proof can be given in the case of real-valued continuous matrix functions under the additional assumption that no accumulation of critical points, i.e., points where the constant rank assumption is not fulfilled, occur. In the following, we will exclude the case that accumulation of critical points occur from our examinations. As a consequence, we can transform to the global canonical form (2.13) on each component Ij separately, but the theory does not allow to treat jumps in the index and in the characteristic values between the intervals Ij of (2.38). Even within the frame-work of impulsive smooth distributions it is not straightforward to define impulsive smooth distributions with impulses allowed at every point of a set
T=I\[
j∈N
Ij
as the set T does not need to be countable. Further, jumps in the characteristic values may affect the solvability within the set of impulsive smooth distributions as can be seen in the following example.
Example 2.52. [82] Consider the initial value problem tx = 0, x(0) = 0.
For this DAE the strangeness index is not defined, but there exits a unique smooth solution of the initial value problem in C1(R,R), namely x = 0. Within the solution space Cimp a possible decomposition according to Corollary 2.51 is given by
R= (−∞,0)∪(0,∞).
Obviously, all distributions of the form x=cδ with c∈R solve the initial value problem.
Hence, we may loose unique solvability when we turn to distributional solutions. Moreover, there is no initial condition of the form (2.31) that fixes a unique solution.
A further drawback of the distributional solution theory presented in Section 2.2.3 is that we have to require infinitely often differentiable matrix-valued functions E(t) and A(t).
Another distributional solution theory for linear DAEs considering so-called piecewise-smooth distributions is presented in [144] that also allows discontinuities in the coefficient matricesE(t) and A(t). In this case, a suitable multiplication for distributions need to be defined. Note, that the space of impulsive smooth distributions is a subspace of piecewise smooth distributions where jumps and Dirac impulse can only occur at timesτi.
Higher Order Differential-Algebraic Systems
General nonlineark-th order differential-algebraic systems of the form
F(t, x,x, . . . , x˙ (k)) = 0, (3.1) with F :I×Dx×Dx˙ × · · · ×Dx(k) →Rm sufficiently smooth on a compact intervalI⊆ R and open sets Dx,Dx˙, . . . ,Dx(k) ⊆ Rn, as well as linear k-th order differential-algebraic equations of the form
Ak(t)x(k)+Ak−1(t)x(k−1)+· · ·+A0(t)x=f(t), (3.2) whereAi ∈C(I,Rm×n) fori= 0,1, . . . , k,k ∈N0andf ∈C(I,Rm) naturally arise in many technical applications. In particular, second order differential-algebraic systems withk = 2 play a key role in the modeling and simulation of constrained dynamical systems, e.g., in the simulation of mechanical multibody systems or in electrical circuit simulation, as we have seen in Chapter 1.
In the classical theory of differential equations, higher order systems are turned into first order systems by introducing new variables for the derivatives. For DAEs this classical approach has to be performed with great care since it may lead to a number of mathematical difficulties as has been discussed in several publications, see [4, 32, 102, 129, 135]. In [102, 135] several examples are presented that show that the classical approach of introducing the derivatives of the unknown vector-valued function x(t) as new variables may lead to higher smoothness requirements for the inhomogeneity f(t) that are needed to ensure the existence of a solution, which corresponds to an increase in the index of the DAE. By introducing only some new variables, however, this difficulty can be circumvented.
Example 3.1. [102] Consider the linear second order constant coefficient DAE 1 0
0 0
¨ x+
1 0 0 0
˙ x+
0 1 1 0
x=f(t), t ∈I, (3.3)
where x= [x1, x2]T, and f = [f1, f2]T. System (3.3) has the unique solution x1 =f2,
x2 =f1−f˙2−f¨2,
and hence the minimum requirement for the existence of a continuous solution is thatf1 is continuous and f2 is twice continuously differentiable. Using the classical transformation to first order by introducing
v = [v1, v2]T = [ ˙x1,x˙2]T, y= [v1, v2, x1, x2]T, 31
we get the additional solution components v1 = ˙f2,
v2 = ˙f1−f¨2−f2(3),
and thus,f2 has to be three times continuously differentiable to obtain a continuous solu-tion. If, however, we only introduce v1 = ˙x2, then no extra smoothness requirements are needed.
Another difficulty that arises in practical numerical applications is that the system may be badly scaled and that there are disturbances and perturbations in the data, such that the transformation to first order may lead to very different solutions in the perturbed system.
Example 3.2. [102] Consider the second order system
ǫ1x¨+ǫ2x˙ +ǫ3x=ǫ4f(t), t∈I, (3.4) with coefficients ǫi, i = 1, . . . ,4 of absolute value close to the machine precision and f of norm approximately 1. If we transform (3.4) to first order in the classical way by introducing
y= [y1, y2]T := [ ˙x, x]T, then we obtain the system
ǫ1 0 0 1
˙ y+
ǫ2 ǫ3
−1 0
y=
ǫ4f(t) 0
. (3.5)
For different values of the ǫi, in finite precision arithmetic, we may decide that the system (3.5) has a unique solution, no solutions at all, or is actually underdetermined.
Recently, it has been shown in [129, 152] that the direct discretization of the second order system may yield better numerical results and is able to prevent certain numerical difficul-ties as the failure of numerical methods, see also [4, 17, 151]. Therefore, a proper treatment of higher order differential-algebraic systems requires either the direct numerical solution of the high order system by appropriate numerical methods as proposed in [129, 152], or carefully chosen first order formulations.
Example 3.3. Consider the example of a multibody system M(p, t)¨p = fa(p,p, t)˙ −GT(p, t)λ,
0 = g(p, t).
In such systems it is common practice to derive a first order formulation by just introducing new variables v = ˙p but not the derivative of λ, i.e.,
˙
p = v,
M(p, t) ˙v = fa(p, v, t)−GT(p, t)λ, 0 = g(p, t).
In this way an unnecessary derivative of the Lagrange multiplierλ is avoided.
The theoretical analysis of linear high order differential-algebraic equations of the form (3.2) regarding existence and uniqueness of solutions has been studied in [102, 135], where condensed forms and corresponding invariants under equivalence transformations are de-rived and a definition of the strangeness-index is given. Further, a stepwise index reduction procedure allows to transform the original system to a strangeness-free system that enables an identification of those higher order derivatives of variables that can be replaced to ob-tain a first order formulation without changing the smoothness requirements. However, the computation of these condensed forms is not numerical feasible as it involves the derivatives of computed transformation matrices.
In this chapter, we first give a brief survey of the relevant results for linear second order DAEs obtained in [102, 135] and introduce the characteristic invariants. Then, we present a numerically computable method to determine the strangeness index as well as the char-acteristic invariants using derivative arrays, following the ideas that were presented in Section 2.2.2. An equivalent strangeness-free differential-algebraic system can be obtained from the original system and its higher derivatives that has the same solution behavior as the original DAE. In Section 3.2 the ideas are extended to nonlinear DAEs. Further, in Section 3.3, we discuss first order formulations for linear second order DAEs and present a trimmed first order formulation, see also [21] for further trimmed first order formulations.
The trimmed first order formulation also allows an explicit representation of solutions for linear second order DAEs with constant coefficient matrices. In the following, we restrict to second order systems for ease of representation and since they are most frequently used.
In principle, all ideas can also be extended to arbitrary k-th order systems.
3.1 Linear Second Order Differential-Algebraic Systems
In this section we consider linear second order differential-algebraic equations with variable coefficients of the form
M(t)¨x+C(t) ˙x+K(t)x=f(t), (3.6) where M, C, K ∈C(I,Rm×n) and f ∈C(I,Rm) are sufficiently smooth functions, together with the initial conditions
x(t0) =x0 ∈Rn, x(t˙ 0) = ˙x0 ∈Rn, for t0 ∈I. (3.7) At first, we introduce the condensed forms that have been derived in [102, 135] for linear second order systems (3.6) that allow to describe the characteristic quantities of the DAE.
In Section 3.1.2 these condensed forms and the relationships between the characteristic quantities are used to extract a strangeness-free reduced system using derivative arrays.