Simplify a system of five differential algebraic equations (DAEs) in four state variables to a system of two equations in two state variables.

Create the following system of five DAEs in four state variables x1(t), x2(t), x3(t), and x4(t). The system also contains symbolic parameters a1, a2, a3, a4, b, c, and the function f(t) that are not state variables.

syms x1(t) x2(t) x3(t) x4(t) a1 a2 a3 a4 b c f(t)
eqs = [a1*diff(x1(t),t)+a2*diff(x2(t),t) == b*x4(t),
       a3*diff(x2(t),t)+a4*diff(x3(t),t) == c*x4(t),
       x1(t) == 2*x2(t),
       x4(t) == f(t),
       f(t) == sin(t)];
vars = [x1(t),x2(t),x3(t),x4(t)];

Use reduceRedundancies to eliminate redundant equations and corresponding state variables.

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

newEqs = 
$$\left(\begin{array}{c}{a}_1\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }{x}_1\left(t\right)+\frac{a_2\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }{x}_1\left(t\right)}{2}-b\hspace{0.17em}f\left(t\right)\\ \frac{a_3\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }{x}_1\left(t\right)}{2}+a_4\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }{x}_3\left(t\right)-c\hspace{0.17em}f\left(t\right)\end{array}\right)$$

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

Specify input order of the state variables to choose which variables are being returned when eliminating DAEs.

Create a system of four DAEs in four state variables V_ac(t), V1(t), V2(t), and I(t). The system also contains symbolic parameters L, R, and V0.

syms V_ac(t) V1(t) V2(t) I(t) L R V0
eqs = [V_ac(t) == V1(t) + V2(t),
       V1(t) == I(t)*R,
       V2(t) == L*diff(I(t),t),
       V_ac(t) == V0*cos(t)]

eqs = 
$$\left(\begin{array}{c}{V}_{\mathrm{ac}}\left(t\right)=V_1\left(t\right)+V_2\left(t\right)\\ V_1\left(t\right)=R\hspace{0.17em}\text{I}\left(t\right)\\ V_2\left(t\right)=L\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }\text{I}\left(t\right)\\ V_{\mathrm{ac}}\left(t\right)=V_0\hspace{0.17em}\cos \left(t\right)\end{array}\right)$$

vars = [V_ac(t),I(t),V1(t),V2(t)]

vars = $$\left(\begin{array}{cccc}{V}_{\mathrm{ac}}\left(t\right)& \text{I}\left(t\right)& V_1\left(t\right)& V_2\left(t\right)\end{array}\right)$$

Use reduceRedundancies to eliminate redundant equations and variables. reduceRedundancies prioritizes to keep the state variables in the vector vars starting from the first element.

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

newEqs = 
$$-L\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }\text{I}\left(t\right)-R\hspace{0.17em}\text{I}\left(t\right)+V_0\hspace{0.17em}\cos \left(t\right)$$

newVars = $$\text{I}\left(t\right)$$

Here, reduceRedundancies returns a reduced equation in term of the variable I(t).

When multiple ways of reducing the DAEs exist, specify a different input order of the state variables to choose which variables are being returned. Specify another vector that contains a different order of the state variables. Eliminate the DAEs again.

vars2 = [V_ac(t),V1(t),V2(t),I(t)]

vars2 = $$\left(\begin{array}{cccc}{V}_{\mathrm{ac}}\left(t\right)& V_1\left(t\right)& V_2\left(t\right)& \text{I}\left(t\right)\end{array}\right)$$

[newEqs,newVars] = reduceRedundancies(eqs,vars2)

newEqs = 
$$-\frac{L\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }{V}_1\left(t\right)+R\hspace{0.17em}{V}_1\left(t\right)-R\hspace{0.17em}{V}_0\hspace{0.17em}\cos \left(t\right)}{R}$$

newVars = $$V_1\left(t\right)$$

Here, reduceRedundancies returns a reduced equation in term of the state variable V1(t).

Declare three output arguments when calling reduceRedundancies to simplify a system of equations and return information about the eliminated equations.

Create the following system of five differential algebraic equations (DAEs) in four state variables x1(t), x2(t), x3(t), and x4(t). The system also contains symbolic parameters a1, a2, a3, a4, b, c, and the function f(t) that are not state variables.

syms x1(t) x2(t) x3(t) x4(t) a1 a2 a3 a4 b c f(t)
eqs = [a1*diff(x1(t),t)+a2*diff(x2(t),t) == b*x4(t),
       a3*diff(x2(t),t)+a4*diff(x3(t),t) == c*x4(t),
       x1(t) == 2*x2(t),
       x4(t) == f(t),
       f(t) == sin(t)];
vars = [x1(t),x2(t),x3(t),x4(t)];

Call reduceRedundancies with three output arguments.

[newEqs,newVars,R] = reduceRedundancies(eqs,vars)

newEqs = 
$$\left(\begin{array}{c}{a}_1\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }{x}_1\left(t\right)+\frac{a_2\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }{x}_1\left(t\right)}{2}-b\hspace{0.17em}f\left(t\right)\\ \frac{a_3\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }{x}_1\left(t\right)}{2}+a_4\hspace{0.17em}\frac{\partial }{\partial t}\mathrm{ }{x}_3\left(t\right)-c\hspace{0.17em}f\left(t\right)\end{array}\right)$$

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

R = struct with fields:
      solvedEquations: [2x1 sym]
    constantVariables: [1x2 sym]
    replacedVariables: [1x2 sym]
       otherEquations: [1x1 sym]

The function reduceRedundancies returns information about eliminated equations to R. Here, R is a structure array with four fields.

The solvedEquations field contains the equations that are eliminated by reduceRedundancies. The eliminated equations contain those state variables from vars that do not appear in newEqs. The right side of each eliminated equation is equal to zero.

R1 = R.solvedEquations

R1 = 
$$\left(\begin{array}{c}{x}_1\left(t\right)-2\hspace{0.17em}{x}_2\left(t\right)\\ x_4\left(t\right)-f\left(t\right)\end{array}\right)$$

The constantVariables field contains a matrix with two columns. The first column contains those state variables from vars that reduceRedundancies replaced by constant values. The second column contains the corresponding constant values.

R2 = R.constantVariables

R2 = $$\left(\begin{array}{cc}{x}_4\left(t\right)& f\left(t\right)\end{array}\right)$$

The replacedVariables field contains a matrix with two columns. The first column contains those state variables from vars that reduceRedundancies replaced by expressions in terms of other variables. The second column contains the corresponding values of the eliminated variables.

R3 = R.replacedVariables

R3 = 
$$\left(\begin{array}{cc}{x}_2\left(t\right)& \frac{x_1\left(t\right)}{2}\end{array}\right)$$

The otherEquations field contains those equations from eqs that do not contain any of the state variables vars.

R4 = R.otherEquations

R4 = $$f\left(t\right)-\sin \left(t\right)$$