In mathematics, a differential-algebraic system of equations (DAE) is a system of equations that either contains differential equations and algebraic equations, or is equivalent to such a system.
The set of the solutions of such a system is a differential algebraic variety, and corresponds to an ideal in a differential algebra of differential polynomials.
In the univariate case, a DAE in the variable t can be written as a single equation of the form
| F(x, |
x,t)=0,
x(t)
| x |
=
| dx | |
| dt |
| |||||
|
In practical terms, the distinction between DAEs and ODEs is often that the solution of a DAE system depends on the derivatives of the input signal and not just the signal itself as in the case of ODEs;[3] this issue is commonly encountered in nonlinear systems with hysteresis,[4] such as the Schmitt trigger.[5]
This difference is more clearly visible if the system may be rewritten so that instead of x we consider a pair
(x,y)
| \begin{align}x(t)&=f(x(t),y(t),t),\\0&=g(x(t),y(t),t).\end{align} |
where
x(t)\in\Rn
y(t)\in\Rm
f:\Rn+m+1\to\Rn
g:\Rn+m+1\to\Rm.
A DAE system of this form is called semi-explicit. Every solution of the second half g of the equation defines a unique direction for x via the first half f of the equations, while the direction for y is arbitrary. But not every point (x,y,t) is a solution of g. The variables in x and the first half f of the equations get the attribute differential. The components of y and the second half g of the equations are called the algebraic variables or equations of the system. [The term ''algebraic'' in the context of DAEs only means ''free of derivatives'' and is not related to (abstract) algebra.]
The solution of a DAE consists of two parts, first the search for consistent initial values and second the computation of a trajectory. To find consistent initial values it is often necessary to consider the derivatives of some of the component functions of the DAE. The highest order of a derivative that is necessary for this process is called the differentiation index. The equations derived in computing the index and consistent initial values may also be of use in the computation of the trajectory. A semi-explicit DAE system can be converted to an implicit one by decreasing the differentiation index by one, and vice versa.[6]
The distinction of DAEs to ODEs becomes apparent if some of the dependent variables occur without their derivatives. The vector of dependent variables may then be written as pair
(x,y)
| F\left(x, |
x,y,t\right)=0
x
\Rn
y
\Rm
t
F
n+m
n+m+1
n
As a whole, the set of DAEs is a function
F:\R(2n+m+1)\to\R(n+m).
Initial conditions must be a solution of the system of equations of the form
| F\left(x(t |
0),x(t0),y(t0),t0\right)=0.
The behaviour of a pendulum of length L with center in (0,0) in Cartesian coordinates (x,y) is described by the Euler–Lagrange equations
\begin{align}
| x,& |
| v&=λ |
y-g,\\ x2+y2&=L2, \end{align}
λ
\begin{align}
&&
|
ux+vy&=0, \end{align}
\begin{align}
&&
|
λ(x2+y2)-gy+u2+v2&=0,\\ ⇒ &&L2λ-gy+u2+v2&=0, \end{align}
| |||
| L |
| ||||
| E=\tfrac32gy-\tfrac12L |
2)+gy
To obtain unique derivative values for all dependent variables the last equation was three times differentiated. This gives a differentiation index of 3, which is typical for constrained mechanical systems.
If initial values
(x0,u0)
y=\pm\sqrt{L2-x2}
y\ne0
v=-ux/y
λ=(gy-u2-v2)/L2
\begin{align}
|
x,\\[0.3em] 0&=x2+y2-L2,\\ 0&=ux+vy,\\ 0&=u2-gy+v2+L2λ. \end{align}
(y0,v0)
DAEs also naturally occur in the modelling of circuits with non-linear devices. Modified nodal analysis employing DAEs is used for example in the ubiquitous SPICE family of numeric circuit simulators.[7] Similarly, Fraunhofer's Analog Insydes Mathematica package can be used to derive DAEs from a netlist and then simplify or even solve the equations symbolically in some cases.[8] [9] It is worth noting that the index of a DAE (of a circuit) can be made arbitrarily high by cascading/coupling via capacitors operational amplifiers with positive feedback.
DAE of the form
| \begin{align}x&=f(x,y,t),\\0&=g(x,y,t).\end{align} |
| \begin{align} x&=f(x,y,t)\\ 0&=\partial |
xg(x,y,t)
| x+\partial |
yg(x,y,t)
| y+\partial |
tg(x,y,t), \end{align}
(
|
\det\left(\partialyg(x,y,t)\right)\ne0.
Every sufficiently smooth DAE is almost everywhere reducible to this semi-explicit index-1 form.
Two major problems in solving DAEs are index reduction and consistent initial conditions. Most numerical solvers require ordinary differential equations and algebraic equations of the form
| \begin{align} | dx |
| dt |
&=f\left(x,y,t\right),\\0&=g\left(x,y,t\right).\end{align}
It is a non-trivial task to convert arbitrary DAE systems into ODEs for solution by pure ODE solvers. Techniques which can be employed include Pantelides algorithm and dummy derivative index reduction method. Alternatively, a direct solution of high-index DAEs with inconsistent initial conditions is also possible. This solution approach involves a transformation of the derivative elements through orthogonal collocation on finite elements or direct transcription into algebraic expressions. This allows DAEs of any index to be solved without rearrangement in the open equation form
| \begin{align}0&=f\left( | dx |
| dt |
,x,y,t\right),\\0&=g\left(x,y,t\right).\end{align}
Once the model has been converted to algebraic equation form, it is solvable by large-scale nonlinear programming solvers (see APMonitor).
Several measures of DAEs tractability in terms of numerical methods have developed, such as differentiation index, perturbation index, tractability index, geometric index, and the Kronecker index.[10] [11]
We use the
\Sigma
\Sigma=(\sigmai,j)
fi
xj
(i,j)
\sigmai,j
xj
fi
-infty
xj
fi
For the pendulum DAE above, the variables are
(x1,x2,x3,x4,x5)=(x,y,u,v,λ)
\Sigma= \begin{bmatrix} 1&-&0\bullet&-&-\\ -&1\bullet&-&0&-\\ 0&-&1&-&0\bullet\\ -&0&-&1\bullet&0\\ 0\bullet&0&-&-&- \end{bmatrix}