reduceDAEToODE

Convert system of first-order semilinear differential algebraic equations to equivalent system of differential index 0

Syntax

newEqs = reduceDAEToODE(eqs,vars)

[newEqs,constraintEqs] = reduceDAEToODE(eqs,vars)

[newEqs,constraintEqs,oldIndex] = reduceDAEToODE(eqs,vars)

Description

newEqs = reduceDAEToODE(eqs,vars) converts a high-index system of first-order semilinear algebraic equations eqs to an equivalent system of ordinary differential equations, newEqs. The differential index of the new system is 0, that is, the Jacobian of newEqs with respect to the derivatives of the variables in vars is invertible.

[newEqs,constraintEqs] = reduceDAEToODE(eqs,vars) returns a vector of constraint equations.

[newEqs,constraintEqs,oldIndex] = reduceDAEToODE(eqs,vars) returns the differential index oldIndex of the original system of semilinear DAEs, eqs.

Examples

Convert DAE System to Implicit ODE System

Convert a system of differential algebraic equations (DAEs) to a system of implicit ordinary differential equations (ODEs).

Create the following system of differential algebraic equations. Here, the symbolic functions x(t), y(t), and z(t) represent the state variables of the system. Specify the equations as a vector of symbolic equations and the variables as a vector of symbolic function calls.

syms x(t) y(t) z(t)
eqs = [diff(x,t) + x*diff(y,t) == y;
    x*diff(x, t) + x^2*diff(y) == sin(x);
    x^2 + y^2 == t*z]

eqs(t) = 
$$\left(\begin{array}{c}x\left(t\right)\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }y\left(t\right)+\frac{\partial }{\partial t}\mathrm{ }x\left(t\right)=y\left(t\right)\\ {x\left(t\right)}^2\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }y\left(t\right)+x\left(t\right)\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }x\left(t\right)=\sin \left(x\left(t\right)\right)\\ {x\left(t\right)}^2+{y\left(t\right)}^2=t\hspace{0.17em}z\left(t\right)\end{array}\right)$$

vars = [x(t), y(t), z(t)]

vars = $$\left(\begin{array}{ccc}x\left(t\right)& y\left(t\right)& z\left(t\right)\end{array}\right)$$

Use reduceDAEToODE to rewrite the system so that the differential index is 0.

newEqs = reduceDAEToODE(eqs,vars)

newEqs = 
$$\begin{array}{l}\left(\begin{array}{c}{\sigma }_1-y\left(t\right)+\frac{\partial }{\partial t}\mathrm{ }x\left(t\right)\\ \left(\cos \left(x\left(t\right)\right)-y\left(t\right)\right)\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }x\left(t\right)-{\sigma }_1\\ z\left(t\right)-2\hspace{0.17em}x\left(t\right)\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }x\left(t\right)-2\hspace{0.17em}y\left(t\right)\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }y\left(t\right)+t\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }z\left(t\right)\end{array}\right)\\ \\ \mathrm{where}\\ \\ \mathrm{ }{\sigma }_1=x\left(t\right)\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }y\left(t\right)\end{array}$$

Reduce System and Return More Details

Check if the following DAE system has a low (0 or 1) or high (>1) differential index. If the index is higher than 1, first try to reduce the index by using reduceDAEIndex and then by using reduceDAEToODE.

Create the system of differential algebraic equations. Here, the functions x1(t), x2(t), and x3(t) represent the state variables of the system. The system also contains the functions q1(t), q2(t), and q3(t). These functions do not represent state variables. Specify the equations as a vector of symbolic equations and the variables as a vector of symbolic function calls.

syms x1(t) x2(t) x3(t) q1(t) q2(t) q3(t)
eqs = [diff(x2) == q1 - x1;
       diff(x3) == q2 - 2*x2 - t*(q1-x1);
       q3 - t*x2 - x3];
vars = [x1(t), x2(t), x3(t)];

Use isLowIndexDAE to check the differential index of the system. For this system, isLowIndexDAE returns 0 (false). This means that the differential index of the system is 2 or higher.

tf = isLowIndexDAE(eqs,vars)

tf = logical
   0

Use reduceDAEIndex as your first attempt to rewrite the system so that the differential index is 1. For this system, reduceDAEIndex issues a warning because it cannot reduce the differential index of the system to 0 or 1.

[newEqs,newVars] = reduceDAEIndex(eqs,vars)

Warning: Index of reduced DAEs is larger than 1.

newEqs = 
$$\left(\begin{array}{c}\frac{\partial }{\partial t}\mathrm{ }{x}_2\left(t\right)-q_1\left(t\right)+x_1\left(t\right)\\ \mathrm{Dx3t}\left(t\right)-q_2\left(t\right)+2\hspace{0.17em}{x}_2\left(t\right)+t\hspace{0.17em}\left(q_1\left(t\right)-x_1\left(t\right)\right)\\ q_3\left(t\right)-x_3\left(t\right)-t\hspace{0.17em}{x}_2\left(t\right)\\ \frac{\partial }{\partial t}\mathrm{ }{q}_3\left(t\right)-x_2\left(t\right)-t\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }{x}_2\left(t\right)-\mathrm{Dx3t}\left(t\right)\end{array}\right)$$

newVars = 
$$\left(\begin{array}{c}{x}_1\left(t\right)\\ x_2\left(t\right)\\ x_3\left(t\right)\\ \mathrm{Dx3t}\left(t\right)\end{array}\right)$$

If reduceDAEIndex cannot reduce the semilinear system so that the index is 0 or 1, try using reduceDAEToODE. This function can be much slower, therefore it is not recommended as a first choice. Use the syntax with two output arguments to also return the constraint equations.

[newEqs,constraintEqs] = reduceDAEToODE(eqs,vars)

newEqs = 
$$\left(\begin{array}{c}\frac{\partial }{\partial t}\mathrm{ }{x}_2\left(t\right)-q_1\left(t\right)+x_1\left(t\right)\\ \frac{\partial }{\partial t}\mathrm{ }{x}_3\left(t\right)-q_2\left(t\right)+2\hspace{0.17em}{x}_2\left(t\right)+t\hspace{0.17em}{q}_1\left(t\right)-t\hspace{0.17em}{x}_1\left(t\right)\\ \frac{\partial }{\partial t}\mathrm{ }{x}_1\left(t\right)-\frac{\partial }{\partial t}\mathrm{ }{q}_1\left(t\right)+\frac{{\partial }^2}{\partial t^2}\mathrm{ }{q}_2\left(t\right)-\frac{{\partial }^3}{\partial t^3}\mathrm{ }{q}_3\left(t\right)\end{array}\right)$$

constraintEqs = 
$$\left(\begin{array}{c}\frac{\partial }{\partial t}\mathrm{ }{q}_3\left(t\right)-q_2\left(t\right)+x_2\left(t\right)\\ x_3\left(t\right)-q_3\left(t\right)+t\hspace{0.17em}{x}_2\left(t\right)\\ -\frac{{\partial }^2}{\partial t^2}\mathrm{ }{q}_3\left(t\right)+\frac{\partial }{\partial t}\mathrm{ }{q}_2\left(t\right)-q_1\left(t\right)+x_1\left(t\right)\end{array}\right)$$

Use the syntax with three output arguments to return the new equations, constraint equations, and the differential index of the original system, eqs.

[newEqs,constraintEqs,oldIndex] = reduceDAEToODE(eqs,vars)

newEqs = 
$$\left(\begin{array}{c}\frac{\partial }{\partial t}\mathrm{ }{x}_2\left(t\right)-q_1\left(t\right)+x_1\left(t\right)\\ \frac{\partial }{\partial t}\mathrm{ }{x}_3\left(t\right)-q_2\left(t\right)+2\hspace{0.17em}{x}_2\left(t\right)+t\hspace{0.17em}{q}_1\left(t\right)-t\hspace{0.17em}{x}_1\left(t\right)\\ \frac{\partial }{\partial t}\mathrm{ }{x}_1\left(t\right)-\frac{\partial }{\partial t}\mathrm{ }{q}_1\left(t\right)+\frac{{\partial }^2}{\partial t^2}\mathrm{ }{q}_2\left(t\right)-\frac{{\partial }^3}{\partial t^3}\mathrm{ }{q}_3\left(t\right)\end{array}\right)$$

constraintEqs = 
$$\left(\begin{array}{c}\frac{\partial }{\partial t}\mathrm{ }{q}_3\left(t\right)-q_2\left(t\right)+x_2\left(t\right)\\ x_3\left(t\right)-q_3\left(t\right)+t\hspace{0.17em}{x}_2\left(t\right)\\ -\frac{{\partial }^2}{\partial t^2}\mathrm{ }{q}_3\left(t\right)+\frac{\partial }{\partial t}\mathrm{ }{q}_2\left(t\right)-q_1\left(t\right)+x_1\left(t\right)\end{array}\right)$$

oldIndex = 
3

Input Arguments

eqs — System of first-order semilinear DAEs

vector of symbolic equations | vector of symbolic expressions

System of first-order semilinear DAEs, specified as a vector of symbolic equations or expressions.

vars — State variables

vector of symbolic functions | vector of symbolic function calls

State variables, specified as a vector of symbolic functions or function calls, such as x(t).

Example: [x(t),y(t)] or [x(t);y(t)]

Output Arguments

newEqs — System of implicit ordinary differential equations

column vector of symbolic expressions

System of implicit ordinary differential equations, returned as a column vector of symbolic expressions. The differential index of this system is 0.

constraintEqs — Constraint equations encountered during system reduction

column vector of symbolic expressions

Constraint equations encountered during system reduction, returned as a column vector of symbolic expressions. These expressions depend on the variables vars, but not on their derivatives. The constraints are conserved quantities of the differential equations in newEqs, meaning that the time derivative of each constraint vanishes modulo the equations in newEqs.

You can use these equations to determine consistent initial conditions for the DAE system.

oldIndex — Differential index of original DAE system eqs

integer

Differential index of original DAE system eqs, returned as an integer.

Algorithms

The implementation of reduceDAEToODE is based on Gaussian elimination. This algorithm is more reliable than the Pantelides algorithm used by reduceDAEIndex, but it can be much slower.

Version History

Introduced in R2014b