nonlinear MPC controller (about deviation term)

[Touraj Eslami]

Dear Prof Löfberg

In nonlinear MPC controller, for predicting the system I turned my equation from

Xk=f(Xk-1,Uk-1)

Yk=g(Xk-1)

To this format :

Xk+Np=Fk*Xk-1+Gk*Uk-1+Hk *ΔU

When we perform Taylor series expansion to linearize the system around the trajectory point, we change the states to a linear format with a deviation term (  X_NL=X ̅+∆X, U_NL=U ̅+∆U  and the same for Y)

Here is my question, what should I use for Uk-1 and Xk-1 ? I know they aren’t the actual state but the deviation term, then what is the amount of these terms ? is it related to the cycle time? or I have to neglect them and just use the Hk*ΔU term to optimize the system?

 

All the best,

Touraj

[Johan Löfberg]

They are precisely what you say they are, U(k-1) and X(k-1), previous solution and previous predictions. You simply carry them over from the last sample

[Touraj Eslami]

but when we linearize around the operating point,I believe the U(k-1) and X(k-1) should be linearized and I cannt use their real value, please correct me if I'm wrong. my question is exactly this,then what should I use here? should I assume its a deviation term with a small value or not?

all the best

[Johan Löfberg]

X(k-1) and U(k-1) are the solutions from the previous iteration, which you have saved from the previous solution

[Alessandro Maria Laspina]

Predictions on what U(k-1) and X(k-1) can be depend on the system. What does this trajectory represent? You will have to pick a starting point based on your limits. Alternatively, you could try and group the x(k) and x(k-1) and like u(..) terms together to have a change in x and u, but you still need to know these 2 variables. 

[Johan Löfberg]

As far as I've understood you're trying to use the good'ol "linearize nonlinear MPC along previous computed trajectory", i.e. poor mans nonlinear optimization

Something like this (don't take this as a reference or any kind of truth or correctness, it was my first summer project as a fresh phd student...)

https://www.diva-portal.org/smash/get/diva2:316602/FULLTEXT01.pdf

0. at time k-1: Magically obtain optimal solution U(k-) with associated trajectories X(k-1)

1. at time k: Linearize MPC along previous solution U(k-1) and X(k-1) to compute optimal U(k) and X(k)

2. k = k+1. Goto 1.

[Johan Löfberg]

oops, saw now that it wasn't a follow up question from the orginal poster but an more of an answer. anyway...