具有与时间有关的输入参数的ODE 15s [英] ODE 15s with time dependent input parameters

问题描述

我有一个MATLAB代码可以解决以下类型的大规模ODE系统

I have a MATLAB code that solves a large scale ODE system of following type

     function [F] = myfun(t,C,u)

u这里是时间相关向量，用作函数myfun中的输入.我当前的代码为:-

u here is a time dependent vector used as a input in the function myfun. My current code reads as:-

 u    = zeros(4,1);
 u(1) = 0.1; 
 u(2) = 0.1;
 u(3) = 5.01/36;
 u(4) = 0.1;

C0            = zeros(15*4*20+12,1);
options = odeset('Mass',Mf,'RelTol',1e-5,'AbsTol',1e-5);
[T,C]   = ode15s(@OCFDonecolumnA, [0,5000] ,C0,options,u); 

我想将整个时域分为5个不同的部分，并让u的元素在不同时间采用不同的值(类似于多步函数).考虑到以后我想解决一个优化问题来确定域[0,5000]中每个时间间隔的u值，什么是最好的编码方式.

I would like to have the entire time domain split into 5 different sections and have the elements of u take different values at different times(something like a multiple step function). what would be the best way to code it considering that I later want to solve a optimization problem to determine values of u for each time interval in the domain [0,5000].

谢谢

推荐答案

ode15s希望您的模型函数接受时间的参数t和矢量x(或您命名的C)微分状态变量.由于ode15s不允许我们传递控制输入的另一个向量u，因此我通常将ODE函数ffcn包装在model函数中:

ode15s expects your model function to accept a parameter t for time and a vector x (or C as you named it) of differential state variables. Since ode15s will not let us pass in another vector u of control inputs, I usually wrap the ODE function ffcn inside a model function:

function [t, x] = model(x0, t_seg, u)
    idx_seg = 0;
    function y = ffcn(t, x)
      % simple example of exponential growth
      y(1) = u(idx_seg) * x(1)
    end
    ...
end

在上面，model的参数是初始状态值x0，切换时间矢量t_seg和不同段中控制输入值的矢量u.您可以从ffcn内部看到idx_seg.这使我们可以将模型集成到所有段中(将上面的...替换为以下内容):

In the above, the parameters of model are the initial state values x0, the vector of switching times t_seg, and the vector u of control input values in different segments. You can see that idx_seg is visible from within ffcn. This allows us to integrate the model over all segments (replace ... in the above with the following):

t_start = 0;
t = t_start;
x = x0;
while idx_seg < length(t_seg)
    idx_seg = idx_seg + 1;
    t_end = t_seg(idx_seg);
    [t_sol, x_sol] = ode15s(@ffcn, [t_start, t_end], x(end, :));
    t = [t; t_sol(2 : end)];
    x = [x; x_sol(2 : end, :)];
    t_start = t_end;
end

在循环的第一次迭代中，t_start为0，t_end为第一个开关点.现在，我们使用ode15s仅在此时间间隔内进行积分，而我们的ffcn用u(1)评估ODE.解决方案y_sol和相应的时间点t_sol附加到我们的总体解决方案(t，x).

In the first iteration of the loop, t_start is 0 and t_end is the first switching point. We use ode15s now to only integrate over this interval, and our ffcn evaluates the ODE with u(1). The solution y_sol and the corresponding time points t_sol are appended to our overall solution (t, x).

对于下一次迭代，我们将t_start设置为当前段的末尾，并将新的t_end设置为下一个切换点.您还可以看到t_seg的最后一个元素必须是模拟结束的时间.重要的是，我们将模拟轨迹的当前尾部y(end, :)传递给ode15s作为下一段的初始状态向量.

For the next iteration, we set t_start to the end of the current segment and set the new t_end to the next switching point. You can also see that the last element of t_seg must be the time at which the simulation ends. Importantly, we pass to ode15s the current tail y(end, :) of the simulated trajectory as the initial state vector of the next segment.

总而言之，函数model将使用ode15s逐段模拟模型，并返回总体轨迹y及其时间点t.您可以通过以下示例来说服自己

In summary, the function model will use ode15s to simulate the model segment by segment and return the overall trajectory y and its time points t. You could convince yourself with an example like

[t, x] = model1(1, [4, 6, 12], [0.4, -0.7, 0.3]);
plot(t, x);

对于我的指数增长示例，它应该产生

which for my exponential growth example should yield

为了进行优化运行，您将需要编写目标函数.该目标函数可以将u传递给model，然后通过查看x来计算与u关联的优点.

For your optimization runs, you will need to write an objective function. This objective function can pass u to model, and then calculate the merit associated with u by looking at x.

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