solving differential riccati equation with a boundary condition

Question

Ifunanya 2014-8-18

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https://ww2.mathworks.cn/matlabcentral/answers/151336-solving-differential-riccati-equation-with-a-boundary-condition

评论： Mingze Yin 2021-12-3

采纳的回答： Aykut Satici

i would like to solve a riccati differential equation using matlab

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Esmail Alandoli 2016-11-3

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

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Answer 1

Aykut Satici 2014-8-18

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https://ww2.mathworks.cn/matlabcentral/answers/151336-solving-differential-riccati-equation-with-a-boundary-condition#answer_148973

编辑：Walter Roberson 2016-11-7

在 MATLAB Online 中打开

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

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jalal khodaparast 2019-10-23

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 2021-12-3

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!

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Answer 2

Esmail Alandoli 2016-11-7

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https://ww2.mathworks.cn/matlabcentral/answers/151336-solving-differential-riccati-equation-with-a-boundary-condition#answer_242252

编辑：Walter Roberson 2016-11-7

在 MATLAB Online 中打开

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)