How can we solve a irregular descriptor system?

Suppose that we have a system with . Also we have assume that the pair may not be regular i.e the matrix E is recengular. My question is how can we solve such type of system? Is there any direct code in Matlab?

Answer 1

[Karan Singh]

Hi Pabitra,

From what I understand the system “Ex=Ax+Bu” is a type of differential-algebraic equation (DAE). Where “x(0)=xo” is the initial condition for the state vector “x” at time “0”. Also, the pair(E,A) may not be regular i.e. the matrix E is rectangular.

To solve such systems, you can use numerical methods. In MATLAB, you can use the ode15s solver which is designed to solve stiff differential equations and DAEs. However, you need to convert your system into a form that ode15s can solve.

Here is a simple example of how to use ode15s:

function dx = dae(t,x) 
    E = ... % define your E matrix 
    A = ... % define your A matrix 
    B = ... % define your B matrix 
    u = ... % define your control input 
    dx = E\(A*x + B*u); % compute the derivative 
end 

x0 = ... % initial condition 
tspan = [0, 10]; % time span 
options = odeset('Mass', E); % set the mass matrix 
[t, x] = ode15s(@dae, tspan, x0, options); % solve the system 

It is important to note that the numerical solution of DAEs is a complex topic and the choice of the solver and its settings can greatly influence the accuracy and efficiency of the solution. Therefore, it's recommended to have a good understanding of the numerical methods for DAEs and the specific requirements of your system.

Attached below are some documentation links that you may find helpful:

• Solve Differential Algebraic Equations (DAEs) - MATLAB & Simulink (mathworks.com)
• Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations | SIAM Publications Library
• odes.pdf (mathworks.com)
• Automatic differentiation of algorithms - ScienceDirect

Hope this helps!

Karan Singh Khati