I would like to model an explictly time dependent differential equation in Modelica.
Say I would like to model
Real x(start=1);
der(x) = t;
in Modelica. What is the correct way to access the time variable t?
The time variable in Modelica is called time and it is accessible in any model or block (but not packages, record, connectors or functions).
Also, instead of using the start attribute I suggest using initial equations. So your complete model would look like this:
model FirstOrder
Real x;
initial equation
x = 1;
equation
der(x) = time;
end FirstOrder;
The equations in the initial equation section are only applied to solve for initial values of states. The equation shown above is trivial, but you can do interesting stuff like:
model FirstOrder_IC2
Real x;
initial equation
x*x*x = 3.0*time+7.0;
equation
der(x) = time;
end FirstOrder_IC2;
The point here is that you can use other equations besides ones that directly specify the value of the state. The above initial equation is not "physically" interesting, but mathematically it is because it is both non-linear and time-varying (i.e. sensitive to the start time of the simulation).
Related
I am new to the field of optimization and I need help in the following optimization problem. I have tried to solve it using normal coding to make sure that I got he correct results. However, the results I got are different and I am not sure my way of analysis is correct or not. This is a short description of the problem:
The objective function shown in the picture is used to find the optimal temperature of the insulating system that minimizes the total cost over a given horizon.
[This image provides the mathematical description of the objective function and the constraints] (https://i.stack.imgur.com/yidrO.png)
The data of the problems are as follow:
1-
Problem data:
A=1.07×10^8
h=1
T_ref=87.5
N=20
p1=0.001;
p2=0.0037;
This is the curve I want to obtain
2- Optimization variable:
u_t
3- Model type:
The model is a nonlinear cost function with non-linear constraints and it is solved using non-linear solver SNOPT.
4-The meaning of the symbols in the objective and constrained functions
The optimization is performed over a prediction horizon of N years.
T_ref is The reference temperature.
Represent the degree of polymerization in the kth year.
X_DP Represents the temperature of the insulating system in the kth year.
h is the time step (1 year) of the discrete-time model.
R is the ratio of the load loss at the rated load to the no-load loss.
E is the activation energy.
A is the pre-exponential constant.
beta is a linear coefficient representing the cost due to the decrement of the temperature.
I have developed the source code in MATLAB, this code is used to check if my analysis is correct or not.
I have tried to initialize the Ut value in its increasing or decreasing states so that I can have the curves similar to the original one. [This is the curve I obtained] (https://i.stack.imgur.com/KVv2q.png)
I have tried to simulate the problem using conventional coding without optimization and I got the figure shown above.
close all; clear all;
h=1;
N=20;
a=250;
R=8.314;
A=1.07*10^8;
E=111000;
Tref=87.5;
p1=0.0019;
p2=0.0037;
p3=0.0037;
Utt=[80,80.7894736842105,81.5789473684211,82.3684210526316,83.1578947368421,... % The value of Utt given here represent the temperature increament over a predictive horizon.
83.9473684210526,84.7368421052632,85.5263157894737,86.3157894736842,...
87.1052631578947,87.8947368421053,88.6842105263158,89.4736842105263,...
90.2631578947369,91.0526315789474,91.8421052631579,92.6315789473684,...
93.4210526315790,94.2105263157895,95];
Utt1 = [95,94.2105263157895,93.4210526315790,92.6315789473684,91.8421052631579,... % The value of Utt1 given here represent the temperature decreament over a predictive horizon.
91.0526315789474,90.2631578947369,89.4736842105263,88.6842105263158,...
87.8947368421053,87.1052631578947,86.3157894736842,85.5263157894737,...
84.7368421052632,83.9473684210526,83.1578947368421,82.3684210526316,...
81.5789473684211,80.7894736842105,80];
Ut1=zeros(1,N);
Ut2=zeros(1,N);
Xdp =zeros(N,N);
Xdp(1,1)=1000;
Xdp1 =zeros(N,N);
Xdp1(1,1)=1000;
for L=1:N-1
for k=1:N-1
%vt(k+L)=Ut(k-L+1);
Xdq(k+1,L) =(1/Xdp(k,L))+A*exp((-1*E)/(R*(Utt(k)+273)))*24*365*h;
Xdp(k+1,L)=1/(Xdq(k+1,L));
Xdp(k,L+1)=1/(Xdq(k+1,L));
Xdq1(k+1,L) =(1/Xdp1(k,L))+A*exp((-1*E)/(R*(Utt1(k)+273)))*24*365*h;
Xdp1(k+1,L)=1/(Xdq1(k+1,L));
Xdp1(k,L+1)=1/(Xdq1(k+1,L));
end
end
% MATLAB code
for j =1:N-1
Ut1(j)= -p1*(Utt(j)-Tref);
Ut2(j)= -p2*(Utt1(j)-Tref);
end
sum00=sum(Ut1);
sum01=sum(Ut2);
X1=1./Xdp(:,1);
Xf=1./Xdp(:,20);
Total= table(X1,Xf);
Tdiff =a*(Total.Xf-Total.X1);
X22=1./Xdp1(:,1);
X2f=1./Xdp1(:,20);
Total22= table(X22,X2f);
Tdiff22 =a*(Total22.X2f-Total22.X22);
obj=(sum00+(Tdiff));
ob1 = min(obj);
obj2=sum01+Tdiff22;
ob2 = min(obj2);
plot(Utt,obj,'-o');
hold on
plot(Utt1,obj)
I have a simple model as follows and I intend to calculate the mean of x derivative wrt time(der(x)).
model Average
Real x;
initial equation
x = 2.0;
equation
der(x) = x + 5;
annotation (experiment(StopTime=10, __Dymola_Algorithm="Dassl"));
end Average;
The point is that it is a simplified code of an original one in which "x" is given by a CombiTimeTable. I need to perform several simulations (by several text files as inputs to the CombiTimeTable) that they do not have the same number of rows (different durations). In other words, StopTime is different from case to case e.g. here the StopTime is 10s for this specific case.
Is there any way to use a general equation to calculate the mean of der(x). I would appreciate if someone could help.
Based on the answer of sjoelund.se, I created a model which computes the mean value of der(x) via 2 methods:
using the block Modelica.Blocks.Math.ContinuousMean from the MSL (instead of Mean, as it does not sample)
computing the mean with (x2-x1)/(time2-time1) when the simulation terminates
model Average
import Modelica.Constants.pi;
import Modelica.Utilities.Streams.print;
Real x;
Real time1, time2, x1, x2;
Modelica.Blocks.Math.ContinuousMean continuousMean;
equation
der(x) = sqrt(1+cos(2*pi*2*time))+sin(time)^2;
// Method 1: Continuous mean from MSL
continuousMean.u = der(x);
when terminal() then
print("Continuous mean: "+String( continuousMean.y));
end when;
// Method 2: Mean at end time
when initial() then
x1 = x;
time1 = time;
end when;
when terminal() then
x2 = x;
time2 = time;
print("Computed mean: "+String( (x2-x1)/(time2-time1)));
end when;
annotation (experiment(StopTime=1.0, __Dymola_Algorithm="Dassl"));
end Average;
Both values are printed at simulation end. They give a similar value as the mean post processing command which you can apply in Dymola on plotted signals:
Another possibility is to call the mean signal operator in Dymola after the simulation has ended (maybe in a script which runs your simulation):
DymolaCommands.Plot.signalOperatorValue("der(x)", SignalOperator.ArithmeticMean, 0, 1);
Couldn't you just calculate the mean as (x2-x1)/(time2-time1), given that you want the mean of der(X)?
Use Modelica.Blocks.Math.Mean if you want a numerical solution instead of an analytic. Note that it samples the signal so there will be some numerical errors due to that. In order to improve the accuracy, you need to reduce the period of sampling but doing so increases the time it takes to simulate the system.
If you do not need to use the mean during simulation it's also possible to use post-processing of the result-file.
I apologize in advance for the question in the title not being very clear. I'm trying to train a reinforcement learning policy using tf-agents in which there exists some unobservable stochastic variable that affects the state.
For example, consider the standard CartPole problem, but we add wind where the velocity changes over time. I don't want to train an agent that relies on having observed the wind velocity at each step; I instead want the wind to affect the position and angular velocity of the pole, and the agent to learn to adapt just as it would in the wind-free environment. In this example however, we would need the wind velocity at the current time to be correlated with the wind velocity at the previous time e.g. we wouldn't want the wind velocity to change from 10m/s at time t to -10m/s at time t+1.
The problem I'm trying to solve is how to track the state of the exogenous variable without making it part of the observation spec that gets fed into the neural network when training the agent. Any guidance would be appreciated.
Yes, that is no problem at all. Your environment object (a subclass of PyEnvironment or TFEnvironment) can do whatever you want within it. The observation_spec requirement is only related to the TimeStep that you output in the step and reset methods (more precisely in your implementation of the _step and _reset abstract methods).
Your environment however is completely free to have any additional attributes that you might want (like parameters to control wind generation) and any number of additional methods you like (like methods to generate the wind at this timestep according to self._wind_hyper_params). A quick schematic of your code would look like is below:
class MyCartPole(PyEnvironment):
def __init__(self, ..., wind_HP):
... # self._observation_spec and _action_spec can be left unchanged
self._wind_hyper_params = wind_HP
self._wind_velocity = 0
self._state = ...
def _update_wind_velocity(self):
self._wind_velocity = ...
def factor_in_wind(self):
self.state = ... #update according to wind
def _step(self, action):
... # self._state update computations
self._update_wind_velocity
self.factor_in_wind()
observations = self._state_to_observations()
...
I appreciate if you can answer my questions or provide me with useful resources.
Currently, I am working on a problem that I need to do alternating optimization. So, consider we have two decision variables x and y. In the first step I take the derivative of loss function wrt. x (for fixed y) and update x. On the second step, I need to take the derivative wrt. y. The issue is x is dependent on y implicitly and finding the closed form of cost function in a way to show the dependency of x on y is not feasible, so the gradients of cost function wrt. y are unknown.
1) My first question is whether "autodiff" method in reverse mode used in TensorFlow works for these problems where we do not have an explicit form of cost function wrt to one variable and we need the derivatives? Actually, the value of cost function is known but the dependency on decision variable is unknown via math.
2) From a general view, if I define a node as a "tf.Variable" and have an arbitrary intractable function(intractable via computation by hand) of that variable that evolves through code execution, is it possible to calculate the gradients via "tf.gradients"? If yes, how can I make sure that it is implemented correctly? Can I check it using TensorBoard?
My model is too complicated but a simplified form can be considered in this way: suppose the loss function for my model is L(x). I can code L(x) as a function of "x" during the construction phase in tensorflow. However, I have also another variable "k" that is initialized to zero. The dependency of L(x) on "k" shapes as the code runs so my loss function is L(x,k), actually. And more importantly, "x" is a function of "k" implicitly. (all the optimization is done using GradientDescent). The problem is I do not have L(x,k) as a closed form function but I have the value of L(x,k) at each step. I can use "numerical" methods like FDSA/SPSA but they are not exact. I just need to make sure as you said there is a path between "k" and L(x,k)but I do not know how!
TensorFlow gradients only work when the graph connecting the x and the y when you're computing dy/dx has at least one path which contains only differentiable operations. In general if tf gives you a gradient it is correct (otherwise file a bug, but gradient bugs are rare, since the gradient for all differentiable ops is well tested and the chain rule is fairly easy to apply).
Can you be a little more specific about what your model looks like? You might also want to use eager execution if your forward complication is too weird to express as a fixed dataflow graph.
so I am making a simple simulation of different planets with individual velocity flying around space and orbiting each other.
I plan to simulate their pull on each other by considering each planet as projecting their own "gravity vector field." Each time step I'm going to add the vectors outputted from each planets individual vector field equation (V = -xj + (-yj) or some notation like it) except the one being effected in the calculation, and use the effected planets position as input to the equations.
However this would inaccurate, and does not consider the gravitational pull as continuous and constant. Bow do I calculate the movement of my planets if each is continuously effecting the others?
Thanks!
In addition to what Blender writes about using Newton's equations, you need to consider how you will be integrating over your "acceleration field" (as you call it in the comment to his answer).
The easiest way is to use Euler's Method. The problem with that is it rapidly diverges, but it has the advantage of being easy to code and to be reasonably fast.
If you are looking for better accuracy, and are willing to sacrifice some performance, one of the Runge-Kutta methods (probably RK4) would ordinarily be a good choice. I'll caution you that if your "acceleration field" is dynamic (i.e. it changes over time ... perhaps as a result of planets moving in their orbits) RK4 will be a challenge.
Update (Based on Comment / Question Below):
If you want to calculate the force vector Fi(tn) at some time step tn applied to a specific object i, then you need to compute the force contributed by all of the other objects within your simulation using the equation Blender references. That is for each object, i, you figure out how all of the other objects pull (apply force) and those vectors when summed will be the aggregate force vector applied to i. Algorithmically this looks something like:
for each object i
Fi(tn) = 0
for each object j ≠ i
Fi(tn) = Fi(tn) + G * mi * mj / |pi(tn)-pj(tn)|2
Where pi(tn) and pj(tn) are the positions of objects i and j at time tn respectively and the | | is the standard Euclidean (l2) normal ... i.e. the Euclidean distance between the two objects. Also, G is the gravitational constant.
Euler's Method breaks the simulation into discrete time slices. It looks at the current state and in the case of your example, considers all of the forces applied in aggregate to all of the objects within your simulation and then applies those forces as a constant over the period of the time slice. When using
ai(tn) = Fi(tn)/mi
(ai(tn) = acceleration vector at time tn applied to object i, Fi(tn) is the force vector applied to object i at time tn, and mi is the mass of object i), the force vector (and therefore the acceleration vector) is held constant for the duration of the time slice. In your case, if you really have another method of computing the acceleration, you won't need to compute the force, and can instead directly compute the acceleration. In either event, with the acceleration being held as constant, the position at time tn+1, p(tn+1) and velocity at time tn+1, v(tn+1), of the object will be given by:
pi(tn+1) = 0.5*ai(tn)*(tn+1-tn)2 + vi(tn)*(tn+1-tn)+pi(tn)
vi(tn+1) = ai(tn+1)*(tn+1-tn) + vi(tn)
The RK4 method fits the driver of your system to a 2nd degree polynomial which better approximates its behavior. The details are at the wikipedia site I referenced above, and there are a number of other resources you should be able to locate on the web. The basic idea is that instead of picking a single force value for a particular timeslice, you compute four force vectors at specific times and then fit the force vector to the 2nd degree polynomial. That's fine if your field of force vectors doesn't change between time slices. If you're using gravity to derive the vector field, and the objects which are the gravitational sources move, then you need to compute their positions at each of the four sub-intervals in order compute the force vectors. It can be done, but your performance is going to be quite a bit poorer than using Euler's method. On the plus side, you get more accurate motion of the objects relative to each other. So, it's a challenge in the sense that it's computationally expensive, and it's a bit of a pain to figure out where all the objects are supposed to be for your four samples during the time slice of your iteration.
There is no such thing as "continuous" when dealing with computers, so you'll have to approximate continuity with very small intervals of time.
That being said, why are you using a vector field? What's wrong with Newton?
And the sum of the forces on an object is that above equation. Equate the two and solve for a
So you'll just have to loop over all the objects one by one and find the acceleration on it.