Prepared by

Engineer Rehab Mohamed Saeed

ABSTRACT. Using the dynamic programming method, we solved the optimal control problem of the bilinear system, which is resulted from the system reformulation. We applied the dynamic programming in its two types (the vicinity and power series expansion methods).

The simulation results are also illustrated to verify usefulness of this technique. 1. Introduction

The great attention is given to the bilinear systems due to two reasons the first reason is the mathematical tractability, as the class of multlinear system includes the class of the linear systems. Moreover, the familiar analytical tools of linear systems can be used as advantage in the multilinear case. The Second reason is that due to the numerous applications that give arise to the bilinear models or generally polynomic models ([1]).

Dynamic programming technique is stated and the dynamic equations of the problem are given with the presentation closely related to the Riccati approach in linear quadratic optimization. Comparing the results obtaind using Dynamic programming method with those obtained using Matrix Riccati technique we prove the convergence of the Dynamic programming technique. Bilinear systems are special class of nonlinear systems possessing many of the properties of linear systems. Hence, there is a need for theoretical results sufficiently close to those already applied for linear systems. The design problem of controllers for bilinear system has been studied by numerous authors ([2], [3]).

Most of the obtained results rely on optimization theory, either using quadratic cost or criteria linear in control, specially through the application of Pontryagin's principle leading to bang-bang controls or minimizing

controls or minimizing control time. Obtaining the optimal solution was not easy because of the nonlinearity in bilinear ([4]).

2. Problem Formulation

The dynamic equation of the bilinear system was considered as : ∑

̇

(1)

Where:

x: is nx1 state vector

u: is mx1 control vector

A,D, and Bi(i=1,…………………………,m) are input matrices of appropriate dimensions.

The optimal control problem considered in this work was stated as follows: 1) Cost function or performance index denoted as J(u) in the form: ()

(

)

()

∫ ( ()

()

( ))

(2)

2) A set of first order differential equations which represent a time-invariant control system, known as the state equation:

(

)

()

̇( )

(3)

Where the vector function f(x,u,t) may be either a linear function in form (

)

()

()

(4)

Or bilinear function which is chosen as a subclass of the general one for simplicity in the form as :

(

)(

( )) ( )

(5)

So the general optimal control problem now is to find an optimal control function u*(t) which minimizes the performance index J(u) given in equation (2).

The definition of the used variables can be listed as

a) x(t) is a state vector with dimension nx1 an assumed to be continuous time vector.

b) u(t) is the control input with dimension mx1 and assumed to be piece wise constant function

()

(

)

(6)

Where h is a sub-interval (sampling period) an is given by

(

c)

d)

e)

f)

g)

h)

)

(7)

and N is the number of samples and k is an index=0,1,……………………N-1 F and Q, are assumed to be constant , symmetric ,positive semi-definite matrices of dimension nxn each.

R assumed to be symmetric positive definite matrix known as weighting matrix of dimension mxm, or positive scalar constant known as weighting factor.

A,B are constant matrices of dimension nxn and nxm respectively. B0 is a constant matrix of dimension nxn.

t0,tf are initial and final fixed times respectively, and

(‘) denotes the vector transposition.

3) Methods of solution

As the control vector assumed to be piece-wise vector so the performance index can be written as

()

(

∫ ( ()

()

)

()

( ))

(8)

The Dynamic Programming method is used...