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Designs of Single Neuron Control Systems: Survey ~陳奇中教授演講投影片
1. Direct Adaptive Process Control Based on
Using a Single Neuron Controller:
Survey and Some New Results
陳奇中
Chyi-Tsong Chen
ctchen@fcu.edu.tw
Department of Chemical Engineering
Feng Chia University
Taichung 407, Taiwan
FCU PSE Lab., C.T. Chen
逢甲大學化工系 1
2. Outline
Introduction
The single neuron controller (SNC) and its parameter
tuning algorithm
Direct adaptive control schemes for chemical processes
using SNCs
Some alternative SNC controllers and their parameter
tuning algorithms
Model-based design of SNC control systems
Conclusions
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3. Introduction
Conventional control strategies and limitations
Structure and design methodologies
─ Open-loop control
─ Manual control
─ Suitable for process whose mathematical model is hard to
characterize precisely 3
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4. Closed-loop control system
─ Use system output error to generate control signal
─ Automatic control
─ Widely used algorithm: PID type controller
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5. PID controller for continuous system
⎡ 1 de ( t ) ⎤
∫0 e ( t ) dt + τ
t
u (t ) = k c ⎢e (t ) +
τI dt ⎥
D
⎣ ⎦
PID controller for discrete system
⎡ Ts k τD ⎤
u (k ) = kc ⎢e(k ) + ∑ e (i ) + ( e ( k ) − e ( k − 1 )) ⎥
⎣ τI i=0 Ts ⎦
k c : proportional gain
τ I : integral time constant
τ D : derivative time constant
TS : sampling time
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6. New challenges
─ Extremely nonlinearities
─ Immeasurable disturbances and uncertainties
─ Unknown or imprecisely known dynamics
─ Time-varying parameters
─ Multi-objectives
Modeling problem
─ Controller parameter's tuning problem
─ Control performance degradation
Motivation:
Searching for new approaches for complex process control
Artificial Intelligence (AI)
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8. Introduction to artificial neural networks
Structure of neurons
An artificial neuron
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9. Multilayer feedforward neural network
receive signals transmit output
from external Signal signals to
environment transmission environment 9
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10. Operations of an artificial neural network
1. Training or learning phase
─ use input-output data to update the network parameters
(interconnection weights and thresholds)
2. Recall phase
─ given an input to the trained network and then generate an
output
3. Generalization (prediction) phase
─ given a new (unknown) input to the trained network and then
gives a prediction
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11. Properties (advantages) of MNN
1. It has the ability of approximating any extremely
nonlinear functions.
2. It can adapt and learn the dynamic behavior under
uncertainties and disturbances.
3. It has the ability of fault tolerance since the quantity
and quality information are distributively stored in
the weights and thresholds between neurons.
4. It is suitable to operate in a massive parallel
framework.
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12. Direct adaptive control using a shape-tunable neural
network controller (Chen and Chang, 1996)
What happen when some neurons of the neural network were
broken down?
+
single neuron controller
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13. The single neuron controller (SNC)
and its parameter tuning algorithm
Single neuron controller
a { 1 − exp [− b(e − θ ) ]}
u (t ) = NL( e, p ) =
1 + exp [ − b (e − θ )]
e(t ) process output error, given by e(t ) = yd (t ) − y (t )
p controller parameter vector, defined as p ≡ [a, b, θ ]T
a control output level
b slope (sensitivity factor)
θ bias
e u
e −θ
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17. A SNC-based direct adaptive control scheme
+ e ek uk u
yd y
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18. SNC parameters tuning algorithm
1
─ System performance E (k ) = ( yd − y (k )) 2
2
─ Parameter tuning algorithm (Chen, 2001)
z (k )
p(k + 1) = p(k ) + η e(k )
1 + z ( k )T z ( k )
where
z ( k ) ≡ ∂y ( k ) ∂ p ( k ) = ( ∂y ( k ) ∂u ( k ) ) Φ ( u ( k ) , p ( k ) )
and
Φ ( u ,p ) ≡ ∂ u ∂ p
⎡u 1 ⎛ u ⎞⎛ u⎞ 1 ⎛ u ⎞⎛ u⎞⎤
= ⎢ , a ( e − θ )⎜ 1 − ⎟ ⎜ 1 + ⎟ , − ab ⎜ 1 − ⎟ ⎜ 1 + ⎟ ⎥
⎣a 2 ⎝ a ⎠⎝ a⎠ 2 ⎝ a ⎠⎝ a⎠⎦
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19. Stability of the SNC parameter learning algorithm
Assume z(k ) is bounded
Let 0 < η < 2 ;
the controller parameter vector p converges to its local
optimal p * asymptotically, where NL (0 , p ∗ ) = u d (the
desired control input) and e(p*) = 0 .
For the theoretical and rigorous proof, please refer to Chen
(2001).
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20. A simplified version of the learning algorithm
--- Using system response direction
parameter tuning algorithm (Chen, 2001)
z (k )
p(k + 1) = p(k ) + η e(k )
1 + z ( k )T z ( k )
where
z ( k ) ≡ ∂y ( k ) ∂ p ( k ) = ( ∂y ( k ) ∂u ( k ) )Φ ( u ( k ) , p ( k ) )
system response direction
z ( k ) = sign ( ∂y ( k ) ∂u ( k ) ) Φ ( u ( k ) , p ( k ) )
Φ ( u ,p ) ≡ ∂ u ∂ p
⎡u 1 ⎛ u ⎞⎛ u ⎞ 1 ⎛ u ⎞⎛ u⎞⎤
= ⎢ , a ( e − θ ) ⎜ 1 − ⎟ ⎜ 1 + ⎟ , − ab ⎜ 1 − ⎟ ⎜ 1 + ⎟ ⎥
⎣a 2 ⎝ a ⎠⎝ a⎠ 2 ⎝ a ⎠⎝ a⎠⎦
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21. Example :
Setpoint : yd = 5
p(0) = [ a(0) b(0) θ (0) ] = [1 1 0]
T T
I.C.
Learning rate : η = 0.15
System response direction: sign ( ∂y ( k ) ∂u ( k ) ) = 1 21
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23. u
SNC shape
tuning progress
e
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24. Direct adaptive control schemes for
chemical processes using SNCs
A SNC-based control scheme for large time-delay processes
(Chen, 2001)
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25. A SNC-based control scheme for non-minimum phase processes
(Chen, 2001)
− +
G p ( s) = G p ( s)G p ( s)
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26. A decentralized SNC control scheme for multi-input/multi-
output processes (Chen and Yen, 1998)
• Consider an n × n multivariable process described by
⎡ y1 ( s ) ⎤ ⎡G11 ( s ) G12 ( s ) L G1n ( s ) ⎤ ⎡ u1 ( s) ⎤
⎢ y ( s ) ⎥ ⎢G ( s ) G ( s ) L G2 n ( s )⎥ ⎢u2 ( s) ⎥
⎢ 2 ⎥ = ⎢ 21 22 ⎥⎢ ⎥
⎢ M ⎥ ⎢ M M O M ⎥⎢ M ⎥
⎢ ⎥ ⎢ ⎥⎢ ⎥
⎣ yn ( s )⎦ ⎣Gn1 ( s ) Gn 2 ( s) L Gnn ( s ) ⎦ ⎣un ( s)⎦
• In loop i , the SNC i produces its controller output through the following
nonlinear mapping (Assume loop paring results are: y i ↔ u i )
~ (t ) = a i { 1 − exp [ − bi [ ei (t ) − θ i ] ] }
ui
1 + exp [ − b [ ei (t ) − θ i ] ]
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27. Parameter tuning algorithm (in continuous form) for SNC i
zi
p i (t ) = η i ei
& , i = 1, 2, K , n
1 + zi zi
T
where z i ≡ sign ( ∂ yi ∂ ui ) Φ ( ui , p i ) and
~ ~
Φ ( ui , p i
~ ) ~
≡ ∂ ui ∂p i
~
⎡ ui 1 ⎛ ~
ui ⎞ ⎛ ~ ⎞ 1 ⎛ ~
ui ⎞ ⎛ ~ T
ui ⎞ ⎤
= ⎢ , ai ( ei − θ i )⎜ 1 − ⎟ ⎜ 1 + ui
⎜ ai ⎟ ⎜
⎟
⎟ , − 2 aibi ⎜ 1 − a ⎟ ⎜ 1 + a ⎟ ⎥
⎜ ⎟⎜ ⎟
⎣ ai 2 ⎝ ⎠⎝ ai ⎠ ⎝ i ⎠⎝ i ⎠⎦
A static decoupler for the decentralized SNC control system:
the decoupling gain Dij (i ≠ j ) can be given simply by
Gij ( s )
Dij = − lim
s →0 Gii ( s)
K ij
=−
K ii 27
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29. Some alternative SNC controllers and their
parameter tuning algorithms
A bounded SNC (Chen and Peng, 1999)
For handling with the input constraint of u min ≤ u (t ) ≤ u max ,
a bounded nonlinear controller of the form
1
u (t ) = [ ( 1 + u (t ) ) u max + ( 1 − u (t ) ) u min
~ ~ ]
2
where
~ (t ) = 1 − exp [ − b
u
( e (t ) − θ ) ]
1 + exp [ − b ( e (t ) − θ ) ]
the parameter tuning algorithm for the bias parameter
θ& (t ) = −η b e (t ) ( 1 − u (t ) ) ( 1 + u (t ) ) sign ⎛ ∂ y ⎞
~ ~ ⎜ ⎟
⎜ ∂u ⎟
⎝ ⎠ 29
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30. A SNC for the temperature trajectory control of a batch
process (Chen and Peng, 1998)
• To achieve tight temperature tracking control
Both heating and cooling of the process
unit are necessary
A parametric variable is used to express the two
manipulated variables simultaneously
u (t ) = 0 : maximum cooling and minimum heating
u (t ) = 1 : maximum heating and minimum cooling
The simplified SNC
1
u (t ) =
1 + exp [ − b(e(t ) − θ )]
• Parameter tuning algorithm
&
θ (t ) = −η b u (t ) (1 − u (t )) e(t ) 30
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31. Unsolved Problem ?
Fact:
System performance depends on the initial
SNC controller parameters.
Question:
How to start up SNC systematically?
Model-based SNC control systems
31
32. Model-based design of SNC control systems
SNC control of first-order processes
The typical function
characteristics of the SNC
− e*
θ e*
─ upper/lower limit part ud
─ linear part
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33. Analysis of the SNC closed-loop control system
Case 1: upper/lower part
⎧ a, e >> e*
u (t ) = ⎨
⎩ − a, e << −e*
− e*
Closed-loop dynamics θ e*
ud
⎧ K p a, e >> e*
⎪
τ y + y = ⎨
&
⎪− K p a, e << −e
*
⎩
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34. Case 2: linear part
since
a[1 − exp( b θ )]
e = 0, u = ud =
1 + exp( b θ )
e =θ, u = 0
Approximated linear function − e*
θ e*
⎛ e(t ) ⎞
u (t ) = ud ⎜ 1 − ⎟ ud
⎝ θ ⎠
⎛ 1 − exp(b θ )) ⎞
u d = a⎜
⎜ 1 + exp(b θ ) ⎟⎟
⎝ ⎠
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35. The closed-loop system dynamics in this case can be represented by
⎛ y − y⎞
τ y
& + y = K p u d ⎜1 − d ⎟
⎝ θ ⎠
Let K P ud = yd , we arrive at
⎛ K ud ⎞ ⎛ K ud ⎞
τ y + ⎜1 −
& ⎟ y = ⎜1 − ⎟ yd
⎝ θ ⎠ ⎝ θ ⎠
or
τ ' y + y = yd
&
where τ ' = τ /(1 − yd / θ ) ≡ α τ is the time constant of the closed-loop system
and α = θ /(θ − yd ) is an index regarding the system performance
The value of θ can be given by
α
θ = yd
α − 1 35
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36. Also, from K P ud = yd we have
yd 1 + exp(bθ )
a = >0
K p 1 − exp(bθ )
⎧ K p a, e >> e*
⎪
we obtain from the solution of τ y + y = ⎨
& * that
⎪− K p a, e << −e
⎩
y (t ) −t
= a (1 − e τ )
KP
yd 1 + exp(bθ ) ⎛
⎜1 − e τ ⎞
−t
= ⎟
K P 1 − exp(bθ ) ⎝ ⎠
Let y (t ) t =4τ ' = yd , then the above equation leads to
1 1 ⎛ yd ⎞
b = ln sign⎜
⎜K ⎟ ⎟
θ 2 e 4α −1 ⎝ P⎠
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37. The SNC parameter value setting procedure is summarized
as follows:
• Given a performance factor α , 0 < α < 1, and the desired process’s output
value y d
one can calculate sequentially the values of θ , b and a from
α
θ = yd
α − 1
1 1 ⎛ yd ⎞
b = ln 4α
sign ⎜
⎜K ⎟
⎟
θ 2e − 1 ⎝ P ⎠
y d 1 + exp( b θ )
a =
K p 1 − exp( b θ )
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38. Hard input constraint u ≤ u
set
a=u
yd 1 + exp(bθ )
Thus from a =
K p 1 − exp(bθ )
>0
⎛ y ⎞
⎜ u+ d ⎟
y d 1 + exp(bθ ) −1 ⎜ KP ⎟
we have a=u = and then b= ln
θ ⎜ u − yd ⎟
K P 1 − exp(bθ )
⎜ KP ⎟
⎝ ⎠
y(t ) yd 1 + exp(bθ ) ⎛
⎜1 − e τ ⎞
−t
Together with = ⎟ and under the
KP KP 1 − exp(bθ ) ⎝ ⎠
condition of y (t ) t =4τ ′ = yd ,
we obtain 1 ⎛⎜1 −
yd ⎞
⎟ and θ = α yd
α = − ln⎜
4 ⎝ KP u ⎟
⎠ α −1
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39. Table 1a. SNC parameter settings for yd ≠ 0
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40. Table 1b. SNC parameter settings for the case of yd = 0
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41. Kp
Example 1: First-order system GP (s ) =
τs + 1
Assume yd = 1
CASE 1: Effects of α on system performance ( Kp =1 , τ =1 )
1
α =0.3
system output
0.8 α =0.5
0.6 α =0.7
0.4
0.2
0
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
3
control input
2.5
2
1.5
1
41
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
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42. CASE 2: α = 0.5 (τ ' / τ )
different time constants different process gain
( K p = 1 fixed) ( τ = 1 fixed)
1 1
system output
system output
k p=1
0.8 τp=1 0.8
k p=5
0.6 τp=5 0.6
k p=10
0.4 τp=10 0.4
0.2 0.2
0 0
0 10 20 30 40 0 2 4 6 8 10
1.8 2
control input
control input
1.6
1.5
1.4
1
1.2
0.5
1
0.8 0
0 10 20 30 40 0 2 4 6 8 10
time time 42
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43. CASE 3: Hard input constraint
If the hard input constraint is u ≤ 2
one can calculate the performance
index as α = 0.1733
for the case of K P = 1 and y d = 1
SNC controller parameters
a=2
b = 5.2412
θ = −0.2096
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44. Model-based SNC control of a first-order plus dead-time
processes
• First-order plus dead-time (FOPDT) process with transfer function of
G p ( s ) = G ( s ) exp( −td s )
where
Kp
G(s) =
(τ s + 1)
The feedforward compensator
is designed as G ff ( s ) = − Gd ( s )
G p (s)
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45. Example 2
1 1
Process: GP ( s ) = e −s , Gd ( s ) = e −0.5 s
s + 1 4s + 1
The feedforward controller : G ff ( s ) = − s + 1
4s + 1
Setpoint: y d = 1
Let α = 0.5, the SNC controller parameter vector is set as
p = [a b θ ]T = [ 1.1565 2.6231 − 1]
T
The IMC-PID controller is given by (Brosilow and Joseph, 2001 )
GPID (s) = 0.610[1 + 1 1.24 s + 0.179 s ( 0.090 s + 1 )]
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46. The performance comparison of SNC with the IMC-PID controller
1.5
SNC
IMC-PID
system output 1
0.5
0
0 10 20 30 40 50 60 70
2
control input
1.5
1
0.5
0 10 20 30 40 50 60 70
time
46
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47. A direct adaptive model-based SNC control system
• The presence of process uncertainties and nonlinearities
plant/model mismatch
In this situation, the associated SNC parameter tuning algorithm
should be implemented to update the parameters.
direct adaptive SNC control system
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48. Example 3: SNC control of a nonlinear process
A bioreactor
X = −D X + μ X
&
1
S = D(S f − S ) −
& μX
YX S
P = − D P + (γ μ + β )X
&
μ is the specific growth rate
⎛ P ⎞
μm ⎜
⎜1 − ⎟S
⎝ Pm ⎟
⎠
μ =
S2
Km + S +
Ki
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49. The control objective is to regulate the concentration of cell mass at
its desired value by manipulating the dilution rate
From open loop test, we have the process model
− 20.576 − s
GP ( s ) = e
2. 4 s + 1
and the disturbance model
0.1092
Gd ( s ) = e− s
5.325 s + 1
The feedforward controller
0.262s + 0.1092
Gff ( s ) =
109.56 s + 20.576
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50. Based on the identified model and let α = 0.1,
We have the initial controller parameter as
p(0) = [ a(0) b(0) θ (0) ] = [0.1474 − 6.1644 − 0.111]
T T
Learning rate: η = 0.1
The PI controller set as K c = − 0.07 L g ⋅ h and τ I = 4.5 h
(Henson and Seborg, 1991 )
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53. Model-based SNC predictive control system
N
─ Model : ym (k + 1) = ∑ hi u (k + 1 − i ) Impulse response model
i =1
─ Predictive model : y (k + 1) = ym (k + 1) + [ y (k ) − ym (k )]
ˆ
y (k + 1) = y (k ) + q(k ) + h1Δu (k )
ˆ
N
where q(k ) = ∑ hi Δu (k + 1 − i )
i =2
a { 1 − exp [− b(e − θ ) ]}
Since u (t ) = NL( e, p ) =
1 + exp [ − b (e − θ )]
Δu (k ) = φ p Δp(k ) + φe Δe(k ) and e(k ) = r (k ) − y (k )
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54. Then
y (k + 1) =
ˆ
1
1 + h1φe
[
r (k ) + q (k ) + h1φ p Δp(k ) + h1φe r (k + 1)
− (1+ h1φe )e(k )]
Objective function
1 1 T
J = w1 [r (k + 1) − y (k + 1)] + Δ p(k )W2 Δp(k )
2
ˆ
2 2
−1
∂J ⎡ w h φpφ
2 T
⎤ w1h1φ p
=0 Δp(k ) = ⎢
1 1 p
+ W2 ⎥
∂Δp(k )
⎣ (1 + h1φe ) ⎥ 1 + h1φe
2
⎢ ⎦
⎡ 1 1 ⎤
⎢ (r (k + 1) − r (k ) ) − q (k ) + e(k )⎥
⎣1 + h1φe 1 + h1φe ⎦
“One-step ahead MPC learning algorithm”
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56. Simulation studies (large time delay + plant/model mismatch)
• Actual process
− 1 −9 s 0.5 −30 s
G p (s ) = e and Gd (s ) = e
1.5s + 1 5s + 1
• Process model
− 1.25
G (s ) =
Gm (s ) = G (s )e −td s where
2s + 1 ,
t d = 10
• CASE 1: Disturbance rejection d(s)=1/s
p(0) = [ a(0) b(0) θ (0) ] = [0.9618 − 0.8318 0] , α = 0 .5
T T
• CASE 2: Setpoint change yd = 1 to yd = −1
p(0) = [ a(0) b(0) θ (0) ] = [2.0332 − 0.8318 −1] , α = 0 .5
T T
Sampling time = 0.5
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61. Direct Nonlinear Control Using SNC
Consider the SNC control of integrating process of order n
a[1 − exp(−bφ )]
y (n)
= , φ = yd − y − θ
1 + exp(−bφ )
and let ( θ generator )
n −1
θ = λ1 y + λ2 y + L + λn −1 y
' '' ( n −1)
= ∑ λi y (i )
i =1
where
yd : setpoint φ
y : process output
a, b : controller parameters
θ : designed variable
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62. • Case 1
When φ is large
⎧φ > 0
y (n)
= ±a ⎨
⎩φ < 0
φ
1 n
y = ± at
n!
, if y (0) = 0
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63. • Case 2
When φ is small
ab
≅ φ = ( yd − y − θ )
(n) ab
y
2 2
ab ⎛ n −1
⎞
= ⎜ yd − y − ∑ λi y ( i ) ⎟
2 ⎝ i =1 ⎠ φ
Taking Laplace transformation
Y ( s) 1
=
Yd ( s ) 2 n
s + λn −1s n −1 + L + λ1s + 1
ab
1 2
=
(εs + 1)n , ε= n
ab 63
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64. • Implementation
SNC control of integrating process of order n
Nonlinear controller
called NLC
yd φ a[1 − exp(−bφ )] y (n) 1 y
1 + exp(−bφ ) sn
y
θ
generator
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65. Example: SNC control of integrating process of order 3
⎡ ⎛ ⎛ 2
(i ) ⎞ ⎞
⎤
a ⎢1 − exp⎜ − b⎜ yd − ∑ λi y ⎟ ⎟⎥
⎜ ⎟
⎢ ⎝ ⎝ ⎠ ⎠⎥
y '' ' = ⎣ ⎦
i =0
⎡ ⎛ 2
( i ) ⎞⎤
1 − exp ⎢− b⎜ yd − ∑ λi y ⎟⎥
⎣ ⎝ i =0 ⎠⎦
⎧ 1 2
⎪ y = ± at , φ = yd − y − λ1 y ' − λ2 y '' large & positive
3!
⎪
⎨ Y (s) 1 1
⎪Y ( s ) = 2 3 =
(εs + 1)3
⎪ d s + λ2 s + λ1s + λ0
2
⎩ ab
2
ε =
3
, λ2 = 3ε 2 , λ1 = 3ε
ab
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66. 1
• SNC control of integrating process
s3
1
a=10, b=1 ( ε = 0.5848)
0.9 a=100, b=1 ( ε = 0.2714)
a=1000, b=1 ( ε = 0.1260)
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0 1 2 3 4 5 6 7 8 9 10
Stepoint change has been made at t=1 66
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67. • Implementation to general linear processes
bm s m + L + b1s + b0
G p (s) =
a n s n + L + a1s + a0 , n≥m
e y ( n−m) 1 an s n + L + a1 s + a0 u bm s m + L + b1 s + b0
yd y
s n−m bm s m + L + b1s + b0 an s n + L + a1 s + 0
Controller
Y (s) 1
=
Yd ( s ) (εs + 1)n − m
67
FCU PSE Lab., C.T. Chen
69. Example: modeling error (plant/model mismatch)
s+6
Actual process : G (s) =
s 4 + 10 s 3 + 35s 2 + 50 s + 24
s + 12
Process model : G p ( s ) =
s 4 + 12 s 3 + 51.5s 2 + 93s + 59.0625
1
a= 10, b= 1 ( ε = 0.5848)
0.9
a= 10, b= 1 ( ε = 0.5848)
0.8
0.7 modeling error
0.6
0.5 no modeling error
0.4
0.3
0.2
0.1
0
0 5 10 15
T im e 69
FCU PSE Lab., C.T. Chen
70. • Application to Nonlinear Process Control
⎧ x = f ( x ) + g ( x )u
&
System : ⎨ relative degree = r
⎩ y = h( x )
Let [
T = h L f h L h L L h M η1 η 2 L η n − r
2
f
r −1
f ]
T
&
ξ1 = ξ 2
&
ξ 2 = ξ3
⎧a = Lg L f r −1h
M ⎪
, ⎪
b = Lf h
r
&
ξ r = b( x) + a ( x)u ⎨
⎪
η = q (ξ ,η )
& ⎪qi = L f Tr +i , i = 1,2,L, n − r
⎩
y = ξ1
70
FCU PSE Lab., C.T. Chen
71. a[1 − exp(− bφ )]
Let b( x) + a ( x)u = v =
1 + exp(− bφ )
v − L f h( x )
r
v − b(ξ ,η )
u= =
a (ξ ,η ) r −1
L g L f h( h)
i.e. ,
& = y ( r ) = a[1 − exp(− bφ )]
ξr
1 + exp(− bφ )
Y (s) 1
=
Yd ( s ) (εs + 1)r
Better than input-output linearization technique (A. Henson and E.
Seborg, Nonlinear process control, 1997) by one order
i.e. , 1
(εs + 1)r +1 71
FCU PSE Lab., C.T. Chen
72. Example: Nonlinear Bioreactor
System :
&
X = − DX + μX
& = D (s − S ) − 1 μX
S f
yx s
P = − DP + [γμ + β ]X
&
y=X
where
⎛ P0 ⎞
μ m ⎜1 − ⎟ S
⎜ P ⎟
μ= ⎝ m ⎠
S2
Km + S +
Ki
72
FCU PSE Lab., C.T. Chen
73. ~=y
x = [X P] , u=D ,
T
S y x s
⎡ x1 ⎤
& ⎡ μ x1 ⎤ ⎡ − x ⎤
⎢x ⎥ = ⎢ − ⎥ ⎢ 1
1
&
⎢ 2⎥ ⎢ ~ μ x1 ⎥ + ⎢ s f − x 2 ⎥ u
⎥
⎢ x3 ⎥ ⎢ y ⎥ ⎢ − x ⎥
⎣& ⎦ ⎢ [ γμ
⎣ + β ] x1 ⎥ ⎣
⎦ 3 ⎦
y = h = x1
a(1 − exp(− bφ )) φ = yd − y −θ
So v= = yd − y
1 + exp(− bφ ) ,
( since r =1, θ = 0 )
v − Lf h v − μx1
u= =
Lg h − x1
Y ( s) 1 2
=
Yd ( s ) εs + 1 , ε = ab
73
FCU PSE Lab., C.T. Chen
77. Conclusions
We have surveyed the recent direct adaptive control strategies
developed based on using the SNCs.
Some alternative SNC-based control schemes as well as the
associated convergence properties have been addressed for the
purpose of dealing with diversified process dynamics.
New results on how to start up the SNC systematically have
been presented.
─ No input constraint: the SNC parameter values can be given by
simply assigning a performance index.
─ on the other hand, a SNC parameter settling formula is provided for the
case that there is a hard input constraint involved. 77
FCU PSE Lab., C.T. Chen
78. Extensive simulation results reveal that, with the systematic parameter
setting formula, the pre-specified performance of the SNC control system
can be ensured if the model is perfect.
Under the situation of plant/model mismatch, the SNC parameter
tuning algorithm can provide a more satisfactory control performance
as compared with conventional linear controllers.
Alternative model-based SNC control systems are also
developed.
--- one-step ahead predictive SNC control
--- nonlinear SNC direct control
78
FCU PSE Lab., C.T. Chen
79. Based on its simple structure and
effective algorithms, the proposed SNC-
based control systems present to be a
promising approaches to the direct
adaptive control of chemical processes.
79
FCU PSE Lab., C.T. Chen
