The Optimization Algorithm of Torque Compensation for PMSM Systems with Periodic-Nonlinear Load - Intelligent Computing for Sustainable Energy and Environment - page 337

mathematical programming techniques to solve problem 1. We solve the gradient
formula of the objective function g 0 ( u )on u before solving the original dynamic
system optimal control problem. When the state values, the co-state values, the
objective function values and the corresponding gradient values of dynamic sys-
tem are known, the sequential quadratic programming (see, for example, [15-21])
is applied to obtain the optimal solution.

3.1 Gradient Formulae

Consider the following system, which is known as the corresponding co-state
system:

∂H 0 ( t, [ x , u ,λ 0 ]( t ))
∂ x

T

dλ 0 ( t )
dt

= − (10 a )

Boundary conditions are given as:

λ 0 ( t )=0 ,t>T (10 b )

H 0 is the corresponding Hamiltonian function defined by:

x 1 ( t )) 2 +( x 2 ( t ) x 2 ( t )) 2 +( λ 0 ) T f ( t,x,u ) H 0 ( t,x,u )=( x 1 ( t ) − − (11)

To solve the dynamical system (3), we divide the interval [0 ,T ]into[0 ,h l ],
[ kh l , ( k +1) h l ], k =1 ,...,η 1,[ ηh l ,T ], where η =INT( T/h l ). The state differ-

ential equation in each subinterval is solved individually.
To solve the co-state system (10), we subdivide the interval [0 ,T ]into[ T

−

−

h l ,T ],[ T ηh l ], where η = Int ( T/h l ).

Then, the co-state system (10) can besolvedoneachshorterinterval.

− ( k +1) h l ,T − kh l ], k =1 ,...,η − 1and[0 ,T −

U

and let Δ u be any bounded measurable Theorem 1. Let u be any control in

− h l ,T ] such that Δ u ( t ) ∈ R

r for each t

∈

[0 ,T ]and Δ u ( t )=

function defined in [

0 for all t

∈ − h l , 0), the directional derivative for the function g 0 is:

[

g 0 ( u + εΔ u ) −g 0 ( u )
ε

= dg 0 ( u + εΔ u )
dε

ε =0

Δg 0 ( u ) = lim
ε

→ 0

(12)

= ∂g 0 ( u )
∂ u

Δ u

= T
0

∂H 0 ( t )
∂ u

Δ u ( t ) dt

where H 0 ( t )= H 0 ( t, x , u ). In [7] we can get the proof of this theorem.

3.2 Control Parameterization

Using the control parameterization method (see, [7] and [12]), we can partition
the time interval [0 ,T ]into q subintervals [ τ j− 1 ,τ j ), where j =1 ,...,q .The
control variable can be approximated as:

q

u q ( t )=

σ q,j χ [ τ j− 1 ,τ j ) ( t )

(13)

j =1