solving differential riccati equation with a boundary condition

i would like to solve a riccati differential equation using matlab

[Esmail Alandoli]

see this link. it might be helpful for you https://www.mathworks.com/help/control/ref/care.html

Answer 1

[Aykut Satici]

Since the Riccati equation is a first-order ordinary differential equation, you can do this easily with any of the ODE solvers available in MATLAB such as "ode45", see

http://www.mathworks.com/help/matlab/ref/ode45.html

The trick is to find the solution backwards in time.

As an example, let us consider the following example. Let the Riccati equation be given by

y'(t) = q0 + q1*y(t) + q2*y(t)^2,  
y(tf) = yf

where q0, q2 are non-vanishing constants (these may be nontrivial functions of t, the fact that they are chosen to be constant is just for simplicity). The second line is the boundary condition that at the end time tf, the value of the solution must be yf. I have chosen, in particular, tf = 2 and yf = 1 in the example code below.

function riccatiEquationRunner()

par = [1;2;1];  % q0, q1, and q2
yf = 1;
ti = 0; tf = 2;
opt = odeset('AbsTol',1.0e-07,'RelTol',1.0e-07);
[t,y] = ode45( @riccatiEquation, [tf,ti], yf ,opt, par);

% Visualize
plot(t,y)

end

function dydx = riccatiEquation(x,y,parameters)

q0 = parameters(1);
q1 = parameters(2);
q2 = parameters(3);

dydx = q0 + q1*y + q2*y*y;

end

[Ifunanya]

would this also work for riccati equations of the form "dX/dt = A'X + XA - XB'BX + Q". ..also the lines of code are somewhat unclear. thanks for the reply

[Aykut Satici]

I am sorry for the code being obscured. I have attached the m-file that contains the code, instead.

The answer to your question is yes. You just need to turn the matrix of first-order ordinary differential equations

dX/dt = A'X + XA - XB'BX + Q

into a vector of ordinary differential equations and write these equations in the function

function dydx = riccatiEquation(x,y,parameters)

Therefore, if the matrix X is n-by-n, then you would have a vector of n^2 first-order differential equations. You can do the conversion from matrix to vector using the function "reshape", as described in

http://www.mathworks.com/help/matlab/ref/reshape.html

[jalal khodaparast]

I apply ode45 to the complex differential riccati equation (solution is complex value) but I get unstable fluctuations in the results? Do you know how to solve this problem?

[Mingze Yin]

Thanks so much for this. However, could you help me on how to write a code for solving a Riccati differential equation of the same form as u suggested, only when q1 and q2 are some function of t (instead of being fixed constants)?

so maybe for example:

y'(t) = q0 + q1*y(t) + q2*y(t)^2;

y(tf) = yf;

where q1 = t, q2 = t^2;

Thanks so much!

Answer 2

[Esmail Alandoli]

Hi,

May you guys help me if you can please?

I have problem with the system below for solving the riccati equation for Y infinity. I always get the error of "Unable to solve the specified Riccati equation because the Hamiltonian spectrum is too close to the imaginary axis."

g = 40000;
A = [0 0 1 0; 0 0 0 1; 0 673.07 -35.1667 0; 0 -1023.07 35.1667 0];
B = [0; 0; 61.7325; -61.7325]';
B1 =[0 0 0 0]';
B2 = B;
C1 = [0 0 0 0]';
C2 = [1 1 0 0]';
C = [C1 , C2]
m1 = size(C1,2)
m2 = size(C2,2)
R = [-g^2*eye(m1) zeros(m1,m2) ; zeros(m2,m1) eye(m2)]
Y = care(A,C,B'*B,R)

can you please help?

Thank you so much

Esmail

[Sana SAAD]

hello, did you find an answer i am in the same issue .