Differential-algebraic system of equations

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 ${\displaystyle (x,y)}$  and the system of differential equations of the DAE appears in the form

${\displaystyle F\left({\dot {x}},x,y,t\right)=0}$ 

where

• ${\displaystyle x}$ , a vector in ${\displaystyle \mathbb {R} ^{n}}$ , are dependent variables for which derivatives are present (differential variables),
• ${\displaystyle y}$ , a vector in ${\displaystyle \mathbb {R} ^{m}}$ , are dependent variables for which no derivatives are present (algebraic variables),
• ${\displaystyle t}$ , a scalar (usually time) is an independent variable.
• ${\displaystyle F}$  is a vector of ${\displaystyle n+m}$  functions that involve subsets of these ${\displaystyle n+m+1}$  variables and ${\displaystyle n}$  derivatives.

As a whole, the set of DAEs is a function

${\displaystyle F:\mathbb {R} ^{(2n+m+1)}\to \mathbb {R} ^{(n+m)}.}$ 

Initial conditions must be a solution of the system of equations of the form

${\displaystyle F\left({\dot {x}}(t_{0}),\,x(t_{0}),y(t_{0}),t_{0}\right)=0.}$ 

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

${\displaystyle {\begin{aligned}{\dot {x}}&=u,&{\dot {y}}&=v,\\{\dot {u}}&=\lambda x,&{\dot {v}}&=\lambda y-g,\\x^{2}+y^{2}&=L^{2},\end{aligned}}}$ 

where ${\displaystyle \lambda }$  is a Lagrange multiplier. The momentum variables u and v should be constrained by the law of conservation of energy and their direction should point along the circle. Neither condition is explicit in those equations. Differentiation of the last equation leads to

${\displaystyle {\begin{aligned}&&{\dot {x}}\,x+{\dot {y}}\,y&=0\\\Rightarrow &&u\,x+v\,y&=0,\end{aligned}}}$ 

restricting the direction of motion to the tangent of the circle. The next derivative of this equation implies

${\displaystyle {\begin{aligned}&&{\dot {u}}\,x+{\dot {v}}\,y+u\,{\dot {x}}+v\,{\dot {y}}&=0,\\\Rightarrow &&\lambda (x^{2}+y^{2})-gy+u^{2}+v^{2}&=0,\\\Rightarrow &&L^{2}\,\lambda -gy+u^{2}+v^{2}&=0,\end{aligned}}}$ 

and the derivative of that last identity simplifies to ${\displaystyle L^{2}{\dot {\lambda }}-3gv=0}$  which implies the conservation of energy since after integration the constant ${\displaystyle E={\tfrac {3}{2}}gy-{\tfrac {1}{2}}L^{2}\lambda ={\frac {1}{2}}(u^{2}+v^{2})+gy}$  is the sum of kinetic and potential energy.

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 ${\displaystyle (x_{0},u_{0})}$  and a sign for y are given, the other variables are determined via ${\displaystyle y=\pm {\sqrt {L^{2}-x^{2}}}}$ , and if ${\displaystyle y\neq 0}$  then ${\displaystyle v=-ux/y}$  and ${\displaystyle \lambda =(gy-u^{2}-v^{2})/L^{2}}$ . To proceed to the next point it is sufficient to get the derivatives of x and u, that is, the system to solve is now

${\displaystyle {\begin{aligned}{\dot {x}}&=u,\\{\dot {u}}&=\lambda x,\\[0.3em]0&=x^{2}+y^{2}-L^{2},\\0&=ux+vy,\\0&=u^{2}-gy+v^{2}+L^{2}\,\lambda .\end{aligned}}}$ 

This is a semi-explicit DAE of index 1. Another set of similar equations may be obtained starting from ${\displaystyle (y_{0},v_{0})}$  and a sign for x.

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.^[4]

DAE of the form

${\displaystyle {\begin{aligned}{\dot {x}}&=f(x,y,t),\\0&=g(x,y,t).\end{aligned}}}$ 

are called semi-explicit. The index-1 property requires that g is solvable for y. In other words, the differentiation index is 1 if by differentiation of the algebraic equations for t an implicit ODE system results,

${\displaystyle {\begin{aligned}{\dot {x}}&=f(x,y,t)\\0&=\partial _{x}g(x,y,t){\dot {x}}+\partial _{y}g(x,y,t){\dot {y}}+\partial _{t}g(x,y,t),\end{aligned}}}$ 

which is solvable for ${\displaystyle ({\dot {x}},\,{\dot {y}})}$  if ${\displaystyle \det \left(\partial _{y}g(x,y,t)\right)\neq 0.}$ 

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

${\displaystyle {\begin{aligned}{\frac {dx}{dt}}&=f\left(x,y,t\right),\\0&=g\left(x,y,t\right).\end{aligned}}}$ 

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

${\displaystyle {\begin{aligned}0&=f\left({\frac {dx}{dt}},x,y,t\right),\\0&=g\left(x,y,t\right).\end{aligned}}}$ 

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 ${\displaystyle \Sigma }$ -method to analyze a DAE. We construct for the DAE a signature matrix ${\displaystyle \Sigma =(\sigma _{i,j})}$ , where each row corresponds to each equation ${\displaystyle f_{i}}$  and each column corresponds to each variable ${\displaystyle x_{j}}$ . The entry in position ${\displaystyle (i,j)}$  is ${\displaystyle \sigma _{i,j}}$ , which denotes the highest order of derivative to which ${\displaystyle x_{j}}$  occurs in ${\displaystyle f_{i}}$ , or ${\displaystyle -\infty }$  if ${\displaystyle x_{j}}$  does not occur in ${\displaystyle f_{i}}$ .

For the pendulum DAE above, the variables are ${\displaystyle (x_{1},x_{2},x_{3},x_{4},x_{5})=(x,y,u,v,\lambda )}$ . The corresponding signature matrix is

${\displaystyle \Sigma ={\begin{bmatrix}1&-&0^{\bullet }&-&-\\-&1^{\bullet }&-&0&-\\0&-&1&-&0^{\bullet }\\-&0&-&1^{\bullet }&0\\0^{\bullet }&0&-&-&-\end{bmatrix}}}$ 

• Algebraic differential equation, a different concept despite the similar name
• Delay differential equation
• Partial differential algebraic equation
• Modelica Language

• http://www.scholarpedia.org/article/Differential-algebraic_equations