We have a stochastic dynamic model: Xk+1 =f(Xk,uk,wk ). We can design a cost function to be optimized using dynamic programming algorithm. How do we design a cost function for this dynamic system to ensure stability?
In Chapter 4 of Ref. [a] for a quadratic cost function and a linear system (Xk+1 =AXk+Buk+wk), a proposition shows that under a few assumptions, the quadratic cost function results in a stable fixed state feedback. However, I think about how we can consider stability issue in the designation of the cost function as a whole when we are going to define the optimal control problem for a nonlinear system generally. Can we use the meaning of stability to design the cost function? Please share me your ideas.
[a] Bertsekas, Dimitri P., et al. Dynamic programming and optimal control. Vol. 1. No. 2. Belmont, MA: Athena scientific, 1995.
optimizati o functie de profit. costul optimizat va da productie zero
Dear Mohammad Reza Salehizadeh
Note that the cost function parameters are selected according to the designer's discretion and desired response speed. But at the same time, this function will be used as the Lyapunov function, and to prove system stability. However we must consider additional constraint for cost function in order to used it as lyapunov function.
In this regard, I suggest you refer to the following book (Appendix B) :
https://sites.engineering.ucsb.edu/~jbraw/mpc/MPC-book-2nd-edition-2nd-printing.pdf
You may have a look at H-infinity Control Theory:
[ https://www.springer.com/gp/book/9781489935618 ], also look at https://www.mit.edu/~dimitrib/Stable_Opt_Control.pdf
https://ieeexplore.ieee.org/abstract/document/50357
Dear Mohammad Reza Salehizadeh
Unfortunately, the attached article
" [a] Bertsekas, Dimitri P., et al. Dynamic programming and optimal control. Vol. 1. No. 2. Belmont, MA: Athena scientific, 1995."
is full of typing errors.
General talking, we consider the linearization of the nonlinear system. Next, we study the stability of the equilibrium state of the new linear system, which indicates the nature of the stability of the nonlinear system in some neighborhoods.
The signs of the real parts of the eigenvalues of the Jacobian matrix decide which approach we should follow. We have the direct and indirect Lyapunov methods to study the stability based on the eigenvalues.
Best regards
Dear Mohammad Reza Salehizadeh,
I suggest you to see links and attached files on topic.
https://www.amazon.fr/Optimal-Control-Estimation-Robert-Stengel/dp/0486682005
https://www.elsevier.com/books/advanced-control-of-chemical-processes-1994/bonvin/978-0-08-042229-9
Best regards
- Badges
- Science topic
- Similar topics
- Engineering
- Control Theory