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Concept
Differential-algebraic system of equations
Summary
In electrical engineering, 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.
In mathematics these are examples of differential algebraic varieties and correspond to ideals in differential polynomial rings (see the article on differential algebra for the algebraic setup).
We can write these differential equations for a dependent vector of variables x in one independent variable t, as
When considering these symbols as functions of a real variable (as is the case in applications in electrical engineering or control theory) we look at as a vector of dependent variables and the system has as many equations, which we consider as functions .
They are distinct from ordinary differential equation (ODE) in that a DAE is not completely solvable for the derivatives of all components of the function x because these may not all appear (i.e. some equations are algebraic); technically the distinction between an implicit ODE system [that may be rendered explicit] and a DAE system is that the Jacobian matrix is a singular matrix for a DAE system. This distinction between ODEs and DAEs is made because DAEs have different characteristics and are generally more difficult to solve.
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; this issue is commonly encountered in nonlinear systems with hysteresis, such as the Schmitt trigger.
This difference is more clearly visible if the system may be rewritten so that instead of x we consider a pair of vectors of dependent variables and the DAE has the form
where , , and
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.
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