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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
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

                                                         2

FCU PSE Lab., C.T. Chen
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

FCU PSE Lab., C.T. Chen
Closed-loop control system




            ─ Use system output error to generate control signal
            ─ Automatic control
            ─ Widely used algorithm: PID type controller


                                                                   4

FCU PSE Lab., C.T. Chen
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
                                                                             5

FCU PSE Lab., C.T. Chen
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)
                                                                     6

FCU PSE Lab., C.T. Chen
Research fields of AI




                               7

FCU PSE Lab., C.T. Chen
Introduction to artificial neural networks
          Structure of neurons




          An artificial neuron




                                                    8

FCU PSE Lab., C.T. Chen
Multilayer feedforward neural network




         receive signals                  transmit output
         from external        Signal      signals to
         environment       transmission   environment 9

FCU PSE Lab., C.T. Chen
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
                                                                         10

FCU PSE Lab., C.T. Chen
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.
                                                                  11

FCU PSE Lab., C.T. Chen
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



                                                               12

FCU PSE Lab., C.T. Chen
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


                                                                               13

FCU PSE Lab., C.T. Chen
The characteristic plots for parameter a




                                                             14

FCU PSE Lab., C.T. Chen
The characteristic plots for parameter b




                                                             15

FCU PSE Lab., C.T. Chen
The characteristic plots for parameter θ




                                                             16

FCU PSE Lab., C.T. Chen
A SNC-based direct adaptive control scheme




           + e            ek   uk     u
   yd                                                y




                                               17

FCU PSE Lab., C.T. Chen
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⎠⎦

                                                                                             18

FCU PSE Lab., C.T. Chen
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).
                                                                 19

FCU PSE Lab., C.T. Chen
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⎠⎦
                                                                                          20

FCU PSE Lab., C.T. Chen
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

FCU PSE Lab., C.T. Chen
Simulation results




                            22

FCU PSE Lab., C.T. Chen
u
      SNC shape
      tuning progress




                              e




                                  23

FCU PSE Lab., C.T. Chen
Direct adaptive control schemes for
           chemical processes using SNCs
        A SNC-based control scheme for large time-delay processes
        (Chen, 2001)




                                                             24

FCU PSE Lab., C.T. Chen
A SNC-based control scheme for non-minimum phase processes
     (Chen, 2001)
                                                         −       +
                                            G p ( s) = G p ( s)G p ( s)




                                                                     25

FCU PSE Lab., C.T. Chen
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 ] ]
                                                                               26

FCU PSE Lab., C.T. Chen
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

FCU PSE Lab., C.T. Chen
A decentralized SNC scheme for 2x2 processes




                                           28

FCU PSE Lab., C.T. Chen
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

FCU PSE Lab., C.T. Chen
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

FCU PSE Lab., C.T. Chen
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
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
                                                        32

FCU PSE Lab., C.T. Chen
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
                                     *
                   ⎩

                                                         33

FCU PSE Lab., C.T. Chen
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 ) ⎟⎟
                    ⎝                ⎠
                                                                   34

FCU PSE Lab., C.T. Chen
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

FCU PSE Lab., C.T. Chen
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⎠

                                                                        36

FCU PSE Lab., C.T. Chen
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 )
                                                                          37

FCU PSE Lab., C.T. Chen
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
                                                                          38

FCU PSE Lab., C.T. Chen
Table 1a. SNC parameter settings for yd ≠ 0




                                                        39

FCU PSE Lab., C.T. Chen
Table 1b. SNC parameter settings for the case of yd = 0




                                                    40

FCU PSE Lab., C.T. Chen
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

FCU PSE Lab., C.T. Chen                                  time
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

FCU PSE Lab., C.T. Chen
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


                                         43

FCU PSE Lab., C.T. Chen
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)




                                                                         44

FCU PSE Lab., C.T. Chen
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 )]

                                                                                      45

FCU PSE Lab., C.T. Chen
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

FCU PSE Lab., C.T. Chen
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




                                                                    47

FCU PSE Lab., C.T. Chen
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

                                                     48

FCU PSE Lab., C.T. Chen
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

                                                                      49

FCU PSE Lab., C.T. Chen
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 )




                                                                     50

FCU PSE Lab., C.T. Chen
Substrate concentration:    +25% variation (150 hr)
                                                       -25% variation (300 hr)
                  8
                                                                                SNC
                 7.5                                                            PI


                  7
            X

                 6.5


                  6
                       0      50    100   150   200   250   300    350    400         450



                0.25

                 0.2

                0.15
            D




                 0.1

                0.05

                  0
                       0      50    100   150   200   250   300    350    400         450
                                                time (hr)                              51

FCU PSE Lab., C.T. Chen
The parameter tuning progress




                                                          52

FCU PSE Lab., C.T. Chen
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 )
                                                                                 53

FCU PSE Lab., C.T. Chen
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”
                                                                                  54

FCU PSE Lab., C.T. Chen
Model-based SNC predictive control of large time-delay
       processes




                                                            55

FCU PSE Lab., C.T. Chen
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
                                                                                 56

FCU PSE Lab., C.T. Chen
Disturbance rejection




                                                  57

FCU PSE Lab., C.T. Chen
58

FCU PSE Lab., C.T. Chen
Setpoint tracking




                                              59

FCU PSE Lab., C.T. Chen
60

FCU PSE Lab., C.T. Chen
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
                                                                            61

FCU PSE Lab., C.T. Chen
• Case 1
       When         φ     is large

                             ⎧φ > 0
         y   (n)
                   = ±a      ⎨
                             ⎩φ < 0
                                                 φ
                        1 n
                   y = ± at
                        n!

                           , if y (0) = 0

                                            62

FCU PSE Lab., C.T. Chen
• 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

FCU PSE Lab., C.T. Chen
• 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




                                                                       64

FCU PSE Lab., C.T. Chen
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
                                                                         65

FCU PSE Lab., C.T. Chen
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

FCU PSE Lab., C.T. Chen
• 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
Example: no modeling error
                                                  s+6
                          G ( s) =
                                     s 4 + 10s 3 + 35s 2 + 50 s + 24
            1
                                                             a=10, b=1 ( ε = 0.5848)
          0.9
                                                             a=10, b=2 ( ε = 0.4642)
          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
                                               Time                                         68

FCU PSE Lab., C.T. Chen
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
• 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
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
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
~=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
closed-loop response for setpoint change
                             6.5



        x1 (biomass conc.)
                              6

                             5.5

                              5

                             4.5
                                   0   1   2        3      4       5       6     7        8   9        10
                                                               time (hr)

                             1.2
                              1                                            the proposed
                                                                           IO
        dilution rate




                             0.8
                                                                           NIMC
                             0.6
                             0.4
                             0.2
                              0
                                   0   1   2        3      4       5       6     7        8   9        10
                                                               time (hr)                          74

FCU PSE Lab., C.T. Chen
Closed-loop response for -20% Y disturbance
                           6.05


      x1 (biomass conc.)     6


                           5.95


                            5.9
                                  0   10     20       30       40       50       60         70        80
                                                            time (hr)

                           0.22

                            0.2                                              the proposed
      dilution rate




                                                                             IO
                           0.18                                              NIMC
                           0.16

                           0.14

                           0.12
                                  0   10     20       30       40       50       60         70        80
                                                            time (hr)                            75

FCU PSE Lab., C.T. Chen
Closed-loop response in the presence of measurement noise




                x1 (biomass conc.)
                                      6

                                     5.9

                                     5.8
                                           0   5     10     15     20     25     30     35     40     45   50
                               0.35
         dilution rate




                               0.25

                               0.15

                               0.05
                                           0   5     10     15     20     25     30     35     40     45   50

                               0.01
      noise signal




                                      0

                         -0.01

                                           0   5     10     15     20      25     30    35     40     45   50
                                                                        time (hr)
                                                                                                           76

FCU PSE Lab., C.T. Chen
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
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
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
Thanks for your attention.

                             80

FCU PSE Lab., C.T. Chen

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  • 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 2 FCU PSE Lab., C.T. Chen
  • 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 FCU PSE Lab., C.T. Chen
  • 4. Closed-loop control system ─ Use system output error to generate control signal ─ Automatic control ─ Widely used algorithm: PID type controller 4 FCU PSE Lab., C.T. Chen
  • 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 5 FCU PSE Lab., C.T. Chen
  • 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) 6 FCU PSE Lab., C.T. Chen
  • 7. Research fields of AI 7 FCU PSE Lab., C.T. Chen
  • 8. Introduction to artificial neural networks Structure of neurons An artificial neuron 8 FCU PSE Lab., C.T. Chen
  • 9. Multilayer feedforward neural network receive signals transmit output from external Signal signals to environment transmission environment 9 FCU PSE Lab., C.T. Chen
  • 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 10 FCU PSE Lab., C.T. Chen
  • 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. 11 FCU PSE Lab., C.T. Chen
  • 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 12 FCU PSE Lab., C.T. Chen
  • 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 13 FCU PSE Lab., C.T. Chen
  • 14. The characteristic plots for parameter a 14 FCU PSE Lab., C.T. Chen
  • 15. The characteristic plots for parameter b 15 FCU PSE Lab., C.T. Chen
  • 16. The characteristic plots for parameter θ 16 FCU PSE Lab., C.T. Chen
  • 17. A SNC-based direct adaptive control scheme + e ek uk u yd y 17 FCU PSE Lab., C.T. Chen
  • 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⎠⎦ 18 FCU PSE Lab., C.T. Chen
  • 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). 19 FCU PSE Lab., C.T. Chen
  • 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⎠⎦ 20 FCU PSE Lab., C.T. Chen
  • 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 FCU PSE Lab., C.T. Chen
  • 22. Simulation results 22 FCU PSE Lab., C.T. Chen
  • 23. u SNC shape tuning progress e 23 FCU PSE Lab., C.T. Chen
  • 24. Direct adaptive control schemes for chemical processes using SNCs A SNC-based control scheme for large time-delay processes (Chen, 2001) 24 FCU PSE Lab., C.T. Chen
  • 25. A SNC-based control scheme for non-minimum phase processes (Chen, 2001) − + G p ( s) = G p ( s)G p ( s) 25 FCU PSE Lab., C.T. Chen
  • 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 ] ] 26 FCU PSE Lab., C.T. Chen
  • 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 FCU PSE Lab., C.T. Chen
  • 28. A decentralized SNC scheme for 2x2 processes 28 FCU PSE Lab., C.T. Chen
  • 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 FCU PSE Lab., C.T. Chen
  • 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 FCU PSE Lab., C.T. Chen
  • 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 32 FCU PSE Lab., C.T. Chen
  • 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 * ⎩ 33 FCU PSE Lab., C.T. Chen
  • 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 ) ⎟⎟ ⎝ ⎠ 34 FCU PSE Lab., C.T. Chen
  • 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 FCU PSE Lab., C.T. Chen
  • 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⎠ 36 FCU PSE Lab., C.T. Chen
  • 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 ) 37 FCU PSE Lab., C.T. Chen
  • 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 38 FCU PSE Lab., C.T. Chen
  • 39. Table 1a. SNC parameter settings for yd ≠ 0 39 FCU PSE Lab., C.T. Chen
  • 40. Table 1b. SNC parameter settings for the case of yd = 0 40 FCU PSE Lab., C.T. Chen
  • 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 FCU PSE Lab., C.T. Chen time
  • 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 FCU PSE Lab., C.T. Chen
  • 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 43 FCU PSE Lab., C.T. Chen
  • 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) 44 FCU PSE Lab., C.T. Chen
  • 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 )] 45 FCU PSE Lab., C.T. Chen
  • 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 FCU PSE Lab., C.T. Chen
  • 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 47 FCU PSE Lab., C.T. Chen
  • 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 48 FCU PSE Lab., C.T. Chen
  • 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 49 FCU PSE Lab., C.T. Chen
  • 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 ) 50 FCU PSE Lab., C.T. Chen
  • 51. Substrate concentration: +25% variation (150 hr) -25% variation (300 hr) 8 SNC 7.5 PI 7 X 6.5 6 0 50 100 150 200 250 300 350 400 450 0.25 0.2 0.15 D 0.1 0.05 0 0 50 100 150 200 250 300 350 400 450 time (hr) 51 FCU PSE Lab., C.T. Chen
  • 52. The parameter tuning progress 52 FCU PSE Lab., C.T. Chen
  • 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 ) 53 FCU PSE Lab., C.T. Chen
  • 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” 54 FCU PSE Lab., C.T. Chen
  • 55. Model-based SNC predictive control of large time-delay processes 55 FCU PSE Lab., C.T. Chen
  • 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 56 FCU PSE Lab., C.T. Chen
  • 57. Disturbance rejection 57 FCU PSE Lab., C.T. Chen
  • 58. 58 FCU PSE Lab., C.T. Chen
  • 59. Setpoint tracking 59 FCU PSE Lab., C.T. Chen
  • 60. 60 FCU PSE Lab., C.T. Chen
  • 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 61 FCU PSE Lab., C.T. Chen
  • 62. • Case 1 When φ is large ⎧φ > 0 y (n) = ±a ⎨ ⎩φ < 0 φ 1 n y = ± at n! , if y (0) = 0 62 FCU PSE Lab., C.T. Chen
  • 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 FCU PSE Lab., C.T. Chen
  • 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 64 FCU PSE Lab., C.T. Chen
  • 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 65 FCU PSE Lab., C.T. Chen
  • 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 FCU PSE Lab., C.T. Chen
  • 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
  • 68. Example: no modeling error s+6 G ( s) = s 4 + 10s 3 + 35s 2 + 50 s + 24 1 a=10, b=1 ( ε = 0.5848) 0.9 a=10, b=2 ( ε = 0.4642) 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 Time 68 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
  • 74. closed-loop response for setpoint change 6.5 x1 (biomass conc.) 6 5.5 5 4.5 0 1 2 3 4 5 6 7 8 9 10 time (hr) 1.2 1 the proposed IO dilution rate 0.8 NIMC 0.6 0.4 0.2 0 0 1 2 3 4 5 6 7 8 9 10 time (hr) 74 FCU PSE Lab., C.T. Chen
  • 75. Closed-loop response for -20% Y disturbance 6.05 x1 (biomass conc.) 6 5.95 5.9 0 10 20 30 40 50 60 70 80 time (hr) 0.22 0.2 the proposed dilution rate IO 0.18 NIMC 0.16 0.14 0.12 0 10 20 30 40 50 60 70 80 time (hr) 75 FCU PSE Lab., C.T. Chen
  • 76. Closed-loop response in the presence of measurement noise x1 (biomass conc.) 6 5.9 5.8 0 5 10 15 20 25 30 35 40 45 50 0.35 dilution rate 0.25 0.15 0.05 0 5 10 15 20 25 30 35 40 45 50 0.01 noise signal 0 -0.01 0 5 10 15 20 25 30 35 40 45 50 time (hr) 76 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
  • 80. Thanks for your attention. 80 FCU PSE Lab., C.T. Chen