Analysis on Data-Based Integrated Learning Control for Batch Processes - Intelligent Computing for Sustainable Energy and Environment - page 133

( )

( )

( )



ς kt ,

e , w k t , ≥ 0

i



i

( )

(13)

wkt , = i = 1, 2, 3



i

( )

( )

( )

ς kt ,

− e w k t , < 0

 

i

i

( )

where is a real number.

By using Lyapunov method [19, 20], the adjustment algorithm of parameters can
be obtained .

ς kt ,

i

3 Performance Analysis

3.1 Convergence Analysis

For the convenience of discussion, it defines that C is an optimization controller
(ILC) of the batch-axis, C is a feedback controller (SNPC) of the time-axis, and G
denotes the batch process.

Theorem 1: If the feedback controller satisfies the condition of (14), the proposed
integrated learning optimization control policy converges with respect to the batch
number k , namely

U Δ→ 0

as k →∞ .

k

( ) ( )

1 + Gj C ωω ≥ 1

(14)

2

Proof: By using the condition of (14), we have
( )
( )

Ej

E ω

1

2

= ≤ 1

( ) ( )

ω 1 + Gj C j ωω

1 2

( ) ( )

GC + C y 1 − CG y

( )

1 2 d 1 d

where and Ey =− =

EyCGy =− =− 1 CGy

1 d 1 d 1 d

2 d

1 + GC 1 + GC

2 2

( ) ( )

The conclusion of means that the tracking error of the inte-

grated optimization control system is less than or equals to that of the system without
real-time feedback control.
Therefore, the time-axis feedback controller satisfies the following inequality

Ej ω ≤ Ej ω

2 1

() ( )

e U ≤ e U (15)

k ILC k ,

Q

Q

Similar to most new controller design methods developed in the literature, perfect
model assumption is assumed in this work in order to develop the first of its kind that
guarantees the convergence of control policy with the proposed integrated control
scheme derived from a rigorous proof. As a result, (7) can be simplified as

( ) ( )

2

2

(16)

min J U , k += 1 e U + U − R U

ILCk , + 1 ILCk , + 1 ILCk , + 1 k

Q

( )

( )

where

e U

+ =− +

y y k 1, t

.

ILC k , 1 d ILC f