Ricatti: applying control signal forward in time

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Chris Dizon
Chris Dizon 2020-3-28
Hey all,
In an optimization class where we are learning about tracking to a desired output.
System
The system is a dc electric motor attached to a rotational load, with the s.s. described by:
x = [x1;x2;x3] = ang pos, vel, current || dx = [dx1;dx2;dx3] || A = [0 1 0; 0 0 4.438; 0 -12 -24] || B= [0;0;20]
There is a performance index, which I won't write, but R=0.01 and Q=[10 0 0;0 0 0;0 0 0]
output y is ang pos, tracking desired y_d(t) = 1
_________________________________________________________________________________________________________________
I was just auditing the class last term and am working through some HW's and have been stuck on applying the control signal forward in time.
I am NOT using the lqr function here; that yieds the steady state answers when T=infinity in the performance index. Here T=10, so the ricatti coefficients and co-state equations are solved first backwards in time, then applied to the control signal forward in time.
Here's what I've done (i.e. ^(T)=transpose):
  • Matlab I solved for the P matrix (defined as dP = PBR^(-1)B^(T)P - PA - A^(T)P - Q || P(T) = 0 )
  • Matlab I solved for the co state matrix (defined as db = -(A - BR^(-1)B^(T)P)^(T)*b + Q*x_d || b(T) = 0 )
  • I made a Simulink model which used gains/sums/integrators/froms/goto blocks to find p1,p2,...p6 and b1,b2,b3 from the above matrices
  • These succesfully gave me the steady state values (when looking at last of data set) of P from using [K,P,sig]=lqr(A,B,Q,R)
  • I also got the correct values for the co-state eq when using the Leibniz formula
  • Basically ^^^^ P and b are verified as correct!
What we know about u:
The optimal control at least, in a tracking problem, is defined as u*(t) = -R^(-1)B^(T)P(t)x(t) - R^(-1)B^(T)b(t)
You can see that the control signal changes in time due to P and b depending on time
In Matlab I managed to get a symbolic array of u*(t) with variables of x1,x2,x3 (since p1...p6 and b1..b3 are already known at each time step).
Problem:
Usually, I would go about this the same way as solving a s.s.
Using the s.s. equation I defined above, use gains/sums/integrators/froms/goto blocks and be fine. But that was when the control signal was constant because p1...p6 were constants.
Here they are changing with time.
I at first used a Matlab function block, where the function is simply y=u_opt_final (this contains the array of changing optimal control signals with time)
However it is not working:
Caused by:
  • Error in default port dimensions function of S-function 'a_control_out/MATLAB Function'. This function does not fully set the dimensions of output port 2
Additionally, Simulink wouldn't know how to step through an array for each time step in its simulation, anyway...
As you can see, I'm sort of stuck :/
I'm attaching my code and the model...if anyone would like to take an in-depth look at it.
OR, if anyone has better advice on how to go about this, I'm all ears. This is my first crack, there is no solution, and I haven't really seen anything online for the past week I have been working on it. At least, what I've seen is the steady state values, but again, that is not what I'm trying to do.
Thanks! That was a lot of info, and I hope any of it made sense, I am happy to clarify.
a_coefficients is parta of HW, where p1..p6 and b1..b3 are solved
a_control_out is where I am currently in a pickle
c_control_out is partc where T=inifinty, and the control signal is known b/c everything is constants. This is just to compare of a system that works
HW3.m is the main file
Chris

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