There is a nonlinear system which name is 'lect7'(lect7).
I Use Matlab commands trim and linmod to find the operating point (OP) and linearize the system model at that OP (My sys)
I'm interested to know is there any way to get system matrix (A,B,C,D) online and at any cycle in matlab simulink?(I mean in my nonlinear system the Gain (0.707) can changes with time so matrix (A,B,C,D) which obtain using this way doesn't have accuracy when times goes.Is there any way to get these matrix at any time and get the accuracy error between nonlinear model and approximated linear system in simulink online and at every cycle?)
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Are there any codes in matlab to to calculate the gains of state-space matrix for nonlinear system? the system equations are written in differential equations and I used Runge-Kutta 4th Order Method to get the dynamic response of the states with time.
Is there any way to estimate the extrapolation using kriging or Gaussian processes regression ?
Gaussian processes work very well for interpolation of scattered data; however, I need to extrapolate a time series of variable in time.
hoe can I extrapolate the x(n+1)
using the history of x variable, x_i , i = n, n-1 ,....
flag
for example, in python: scikit-learn.org/stable/modules/gaussian_process.html
Extrapolation works in the same way theoretically and practically.
In theory, when you learn a Gaussian process regression model, you have modelled a Gaussian process on your data, you selected its mean function, its covariance function and have estimated their parameters. To interpolate (or extrapolate), you compute the mean of this Gaussian process at a new point, knowing the learning points.
In practice, for both interpolation and extrapolation, you just have to call a prediction function (called predict in R package DiceKriging and in scikit-learn in python).
However, you must known that Gaussian process regression (as many of the regression techniques [citation needed] works quite bad in extrapolation. The Gaussian process mean quickly "returns" to the function mean you have defined. Then, Gaussian process regression in extrapolation is just parametric regression whose model is the one you have chosen for the mean function.
Hopefully, I will be able to explain my question well.
I am working on Nonlinear model predictive control implementation.
I have got 3 files:
1). a simulink slx file which is basically a nonlinear pendulum model.
2). A function file, to get the cost function from the simulink model.
3). MPC code.
code snippet of cost function
**simOut=sim('NonlinearPendulum','StopTime', num2str(Np*Ts));**
%Linearly interpolates X to obtain sampled output states at time instants.
T=simOut.get('Tsim');
X=simOut.get('xsim');
xt=interp1(T,X,linspace(0,Np*Ts,Np+1))';
U=U(1:Nu);
%Quadratic cost function
R=0.01;
J=sum(sum((xt-repmat(r,[1 Np+1])).*(xt-repmat(r,[1 Np+1]))))+R*(U-ur)*...
(U-ur)';
Now I take this cost function and optimize it using fmincon to generate a sequence of inputs to be applied to the model, using my MPC code.
A code snippet of my MPC code.
%Constraints -1<=u(t)<=1;
Acons=[eye(Nu,Nu);-eye(Nu,Nu)];
Bcons=[ones(Nu,1);ones(Nu,1)];
options = optimoptions(#fmincon,'Algorithm','active-set','MaxIter',100);
warning off
for a1=1:nf
X=[]; %Prediction output
T=[]; %Prediction time
Xsam=[];
Tsam=[];
%Nonlinear MPC controller
Ubreak=linspace(0,(Np-1)*Ts,Np); %Break points for 1D lookup, used to avoid
% several calls/compilations of simulink model in fmincon.
**J=#(v) pendulumCostFunction(v,x0,ur,r(:,a1),Np,Nu,Ts);**
U=fmincon(J,U0,Acons,Bcons,[],[],[],[],[],options);
%U=fmincon(J,U0,Acons,Bcons);
U0=U;
UUsam=[UUsam;U(1)];%Apply only the first selected input
%Apply the selected input to plant.
Ubreak=[0 Ts]; %Break points for 1D lookup
U=[UUsam(end) UUsam(end)];
**simOut=sim('NonlinearPendulum','StopTime', num2str(Ts));**
In both the codes, I have marked the times we call our simulink model. Now, issue is that to run this whole simulation for just 5 seconds it takes around 7-8 minutes on my windows machine, MATLAB R2014B.
Is there a way to optimize this? As, I am planning to extend this algorithm to 9th order system unlike 2nd order pendulum model.
If, anyone has suggestion on using simulink coder to generate C code:
I have tried that, and the problem I face is that I don't know what to do with the several files generated. Please be as detailed as possible.
From the code snippets, it appears that you are solving a linear time invariant model with a quadratic objective. Here is some MATLAB (and Python) code for an overhead crane pendulum and inverted pendulum, both with state space linear models and quadratic objectives.
One of the ways to make it run faster is to avoid a Simulink interface and a shooting method for solving the MPC. A simultaneous method with orthogonal collocation on finite elements is faster and also enables higher index DAE model forms if you'd like to use a nonlinear model.
I have just started understanding modeling techniques based on regression models and was going through MATLAB curve fitting toolbox and the SO. I have fundamental doubts and unable to proceed further. I have a single vector set with k=100 data points which I want to fit into an AR model,MA model,ARMA model successively to see which is better suited.Starting with an AR(p) model of the form y(k+1)=a*y(k)+ b*y(k-1)The command
coeff = polyfit(x,y,d)
will fit a polynomial of degree say d=1 with p number of coefficients indicating the order of the model (AR(p)). But I just have 1 set of data which is the recording of the angular moment.So,what will go as the first parameter (x) of the function signature i.e what will be x,y?Then, what if the linear models are not good enough so I may have to select the nonlinear models.Can somebody please guide with code snippets what are the steps in fitting,checking for overfitting,residual calculation etc.
x is likely to be k (index of y). And the whole code:
c =polyfit(1:length(y), y, d).
Matlab has a curve fitting toolbox. You could use it to check different nonlinear fitting in GUI to get some intuition.
If you want steps there's a great Coursera Machine Learning course. The beginning of this course is related to linear regression and I recommend you to spend some hours at least on that beginning.
Drawing bifurcation diagram for 1D system is clear but if I have 2D system on the following form
dx/dt=f(x,y,r),
dy/dt=g(x,y,r)
And I want to generate a bifurcation diagram in MATLAB for x versus r.
What is the main idea to do that or any hints which could help me?
You first have to do some math:
Setting each of the functions to zero gives you two functions y(x) (called the nullclines), which you can plot in a phase diagram. Where the two lines intersect are the fixed-points (equilibria) of your system.
Now, you have to take the jacobian of your system and plug each of those fixed-points in, which will give you the linear stability analysis of the system.
The location of the fixed points and the stability of each point can now be computed as a you vary r (the bifurcation parameter).
For the programming:
-use newton's method (fsolve in MATLAB) to find where the equations are zero
-eig will help you find the eigenvalues of the system.
However
It depends on your system.
If you're supposed to be looking for limit cycles or chaos or something, you'll have to use one of the ode solvers and then the analysis becomes more tricky. I suppose you could develop a poincare-bendixson algorithm, but that would be involved and details would depend on your system.
I don't think MATLAB has anything built in that would give you a bifurcation diagram. There is this third-party solution:
http://www.mathworks.com/matlabcentral/fileexchange/8382