discretize a nonlinear model in YALMIP

[陈海亮]

hi,Dr.Johan

 i want to ask how to discretize  a nonlinear continuous model  (dx/dt=Ax(t)+Bu(t)+w(t))   in yalmip ?

Thanks

[Johan Löfberg]

If you mean A and B are sdpvars, then you cannot apply c2d. The closest you can do is to use a simple Euler approximation x(t+Ts) = x(t) + Ts*(Ax(t) + Bu(t)) = (I+TsA)x(t) + (TsB)u(t)

[陈海亮]

yes, my A and B are sdpvars, and does 'w' (sdpvar)  keeps unchanged during discretization？

[Johan Löfberg]

same thing of course x(t+Ts) = x(t) + Ts*(Ax(t) + Bu(t)+w(t)).

However, didn't we discuss precisely this thing last week. If you want to create an optimizer object where your parameters influence the continuous-time data but your optimization model involves the discrete-time model, then you parameterize the model in terms of the discrete-time model, and compute the discrete-time model when you have the data, before calling the optimizer object. 

[陈海亮]

yes, i know. i will follow that structure and thanks for your support.

[陈海亮]

Hello,Dr.Johan

 x(t+Ts) = x(t) + Ts*(Ax(t) + Bu(t)+w(t)) , this discretization method seems didn't work.

here is part of my code

sdpvar  Ad,Bd,Edw

        Adr=Ts*Ac+eye(4);

        Bdr=Ts*Bc;

        Edwr=Ts*wr;  (Ac,Bc,wr are continuous model matrice and calculated in advance)

………………

 controller = optimizer(constraints,objective,ops,{Ad,Bd,Edw,x{1},r},u{1});

         uout = controller{{Adr,Bdr,Edwr,currentx,currentr}};

[Johan Löfberg]

didn't work is a pretty vague description...

note that optimizer no longer requires cells when calling, i.e, you can write  controller(Adr,Bdr,Edwr,currentx,currentr)

[Johan Löfberg]

and why are you using this simple discretization when you just as well can use c2d

[陈海亮]

ok, i know.

c2d seems didn't work too. 

But i doubt discretization model lead to the problem  for the reason that i  meet the same situation before which is as follows(a trajectory following control based on MPC,where bule line is the reference trajectory, red one is the real trajectory)

[Johan Löfberg]

obviously it's not about the discretization, but about the general tuning and setup of your controller

[Saeed]

Dear Johan,

When we use Euler backward method, x(k-1) will appear in difference equation as x(t)- x(t-1) =Ts*(Ax(t) + Bu(t)) =>x(t)= (I-TsA)^-1 x(t-1) + Ts(I-TsA)^-1 Bu(t)
So, for simulation, the will  state matrix be (I-TsA)^-1 ? and the input matrix Ts(I-TsA)^-1 B?

Is there any special difference between backward and Forward Euler methods, when we apply the discrete model in MPC simulation?

[Johan Löfberg]

there is a whole plethora of methods for discretizing linear dynamics, with Euler being the trivial but non-exact one. read, e.g., the reference pages on c2d and go from there to learn the theory

[Saeed]

Thanks a lot,
But what about the state matrix in backward euler method?
Ad = (I-TsA)^-1 ??
Should we use x(t)= (I-TsA)^-1 x(t-1) + Ts(I-TsA)^-1 Bu(t)  for prediction model in nominal MPC in YALMIP?

[Johan Löfberg]

In a real situation where you don't need a simple parameterization, you would use an exact zero-hold based on integration of the dynamics, i.e, what you get from c2d. Euler approximation is just a hack to get something easily parameterized in continuous A matrix

[陈海亮]

hi,Dr Johan

sorry to interrupt you again.

 

i want to ask something about discretize a LTI model, dx/dt=Ax(t)+Bu(t)+W, Y=Cx(t)+Du(t),    where A,B,C,D,W are constant.

here i have some methods,will they all be ok？(The main problem is that i don't know how to deal with W)

1.x(k+1)-x(k)=Ts*Ax(k)+Ts*Bx(k)+Ts*W =>x(k+1)=(Ts*A+I)x(k)+Ts*Bx(k)+Ts*W 

2.x(k+1)-x(k)=(Ts*A+I)x(k)+Ts*Bx(k)+W ？？

3.dmodel=c2d(A,B,C,D),Ad=dmodel.A,Bd=dmodel.B =>x(k+1)=Adx(k)+Bdx*u(k)+W

4.dx/dt=Ax(t)+Bu(t)+W=>dx/dt=Ax(t)+Bu(t)+I*W=>dx/dt=Ax(t)+[B I]*[u(t) W]'=>c2d(A,[B I],C,D) （where I is the unit matrix）=>split  discretized matrice [B I] to get the discretize matrice of B,W

Thanks

[Johan Löfberg]

You appear to have a weak understand of what a discretization actually is, otherwise it would be obvious that 2 is clearly wrong and you are treating W strangely

dxdt = f(x,q) is can be approximated by x(t+Ts) = x(t) + Ts*f(x(t),q(t)). Whether q involves control or disturbances is not interesting, its just a simple Euler approximation of the differential equation

When f is linear dxdt = Ax + Bq, you can create an exact zoh prediction x(t+Ts) = Fx(t) + Gq(t). Once again, the fact that q = [u;W] is not interesting. As long as q is an external zoh signal, that is an exact prediction

[陈海亮]

that seems 1 and 4 are right.

Additional question, i know zoh but i don't know why q msut be  an external zoh signal?  i don't understand well.

[Johan Löfberg]

They must be constant during the sample, otherwise it is impossible to solve the differential equation over one sample analytically, which is done when you apply c2d. c2d analytically solves the differential equation dxdt = Ax + Bq for Ts seconds, under assumption that q is constant during this time. Being external is just another way of saying that is doesn't vary as a function of x or something like that. Constant is the only important property

[陈海亮]

But q  involves control in the most situations and control is to be solved , how do i know it is an external zoh signal?

[Johan Löfberg]

Whether God chose the value, you did based on an algorithm, or whatever, does not matter, as long as it is fixed over the sample.

Start reading on the section Derivation.

https://en.wikipedia.org/wiki/Discretization