1. Optimization of Multistage Dynamic Systems
By Yu-Chi Ho and Jonathan T. Lee
Harvard University
Aug. 12, 2000
Version 3.1
Date:

2. Outline Motivation Problem Statement Problem Formulation Solution
Reference

3. Motivation Most of the systems of interest are dynamic systems.
For example: airplane flight, chemical processes, room temperature control and many more.

4. Problem Statement Control at time 0 u0 u1 uN-1 x2 x1 xN-1 x0 f 0 f 1 …
f N-1 xN
The index of the variables is time.
x’s are the state variables.
u’s are the control variables.
f’s are the system dynamics.
State at time 0
System dynamics at time 0 N

5. Problem Statement (cont.)
Find the sequence of ui’s to minimize the performance criterion while obeying the system dynamics xi + 1 = f i[xi, ui], given x0, for i = 0, …, N – 1.

6. Problem Formulation Let and Rewrite the performance criterion as N
The bars over a variable means it is a vector. N

7. Problem Formulation (cont.)
Write the state transition as constraint:

8. Problem Formulation (cont.)
The problem becomes minimizing subject to the constraint
This is a static optimization problem.

9. Solution
Let where
The necessary conditions for the solution are

10. Solution (cont.) The first necessary conditions reduce to
with the boundary condition x0
with the boundary condition N =  / xN.
The above two difference equations are equivalent to the first two necessary conditions in vector form.
This is called a two-point boundary-value problem.
Attention: the selection of lambda is for the convenience of the analysis of du to dh. It just hides the affection of du to dx.

11. Reference
Bryson, Jr., A. E. and Y.-C. Ho, Applied Optimal Control: Optimization, Estimation, and Control, Taylor & Francis, (Sections 1.2, 1.3, 2.1 and 2.2)