Design and analysis of neural/fuzzy variable structural PID control systems

is of n

C.-L. Chen F.-Y.Chang

Indexing terms: NeuraUfuzzy system, Variable structural controller, PID, p H control

Abstract: The paper describes the design method of a neuralifuzzy variable structural proportional- integral-derivative (neurallfuzzy VSPID) control system. The neurallfuzzy VSPID controller has a structure similar to that of the conventional PID. In this controller, the PD mode is used in the case of large errors to speed up response, whereas the PI mode is applied for small error conditions to eliminate the steady-state offset. A sigmoidal-like neuron is employed as a preassigned algorithm of the law of structural change. Meanwhile, the controller parameters would be changed according to local conditions. Bounded neural networks or bounded fuzzy logic systems are used for constructing the nonlinear relationship between the PID controller parameters and local operating control conditions. Flexible changes of controller modes and resilient controller parameters of the neuralifuzzy VSPID during the transient could thereby solve the typical conflict in nature between steady-state error and dynamic responsiveness. A neutralisation process is used to demonstrate the applicability of such a controller for controlling highly nonlinear processes.

The three-mode proportional-integral-derivative (PID) controller is widely used in chemical plants due to ease of use and robustness in the face of plant uncertainties. Nevertheless, the linear PID algorithm might be diffi- cult to deal with processes with complex dynamics, such as those with large dead time, inverse response and highly nonlinear characteristics. To date, many sophisticated algorithms have been used to help the PID controller work under such difficult conditions. The various nonlinear PID controllers in which the simple controller structure has been reserved and supe- rior performance has been achieved by allowing con-

troller parameters to vary with local control conditions, such as three-piece PIDs [I] and nonlinear PI(D)s [2, 31, seem to be acceptable. Meanwhile, neural/fuzzy infer- ences based on self-tuning schemes of PI controllers 0 IEE, 1996

IEE Proceedings online no. 19960261

Paper first received 9th May 1995 and in revised form 20th December 1995

The authors are with the Department of Chemical Engineering, National Taiwan University, Taipei 10617, Taiwan, Republic of China

have also been proposed to improve the control per- formance (for example, see [4, 51) by using the neural/ fuzzy capabilities to store the domain expert knowledge and to infer control decisions. On the other hand, improving the limited performance of PI controllers, such as the conflict in nature between static accuracy (steady-state error) and dynamic responsiveness (speed of response), the variable structural PID (VSPID), i.e. using the PD action to accelerate the speed of the response and using the PI mode to eliminate the steady-state offset, could be used to overcome the diffi- culties. The VSPID controller has a structure which is flexibly changed by a preassigned algorithm of the law of structural change. Its superior performance, with the aid of neural/fuzzy systems, additionally achieved by allowing controller parameters to vary with local con- trol conditions, is proposed in this paper.

Radial basis function networks (RBFNs) and back- propagation neural networks (BPNNs) have yielded useful results in many practical areas such as pattern recognition [6], system identification [7, 81 and control [9], due primarily to their simple structures for realisa- tion and well established training algorithms. Many fuzzy paradigms, meanwhile, have been studied in recent years by viewing a fuzzy logic system (FLS) as a functionally equivalent RBFN or BPNN [lo, 111. As indicated in [l 11, the most important advantage of such an FLS spanned by fuzzy basis functions is the provi- sion of a natural framework for combining numerical values and linguistic symbols in a uniform way. From a mathematical point of view, the input-output expres- sions of those mappings are identical in spite of the dis- tinct inference procedure. Capability discrimination between neural and fuzzy systems is thus diminished for proofs of universal neuralifuzzy approximators [l 1, 121. Using neural networks or fuzzy systems to approx- imate a given plant or to control a process now depends on whether rich available data are at hand or whether the ‘If-Then’ control heuristics could be estab- lished by human experts familiar with system dynamics under consideration. The RBFN, the BPNN and the FLS are used interchangeably in this paper since they could provide equivalent functionality.

The design methodology of a neuralifuzzy variable structural PID controller (neural/fuzzy VSPID) for nonlinear processes is proposed in this paper. The neu- ralifuzzy systems discussed above are exploited to pro- vide the nonlinear VSPID parameters according to local control conditions. A simple sigmoidal-like neu- ron is employed as a preassigned algorithm of the law of structural change which is directed by the current value of the error signal. High-quality control could be IEE Proc -Control Theory A p g l , Vol 143, No 2, March 1996 200

assured if parameters of the neuralifuzzy VSPID are suitably determined and the law of structural change could be deduced properly. The PI mode with antireset windup is designed for the prevention of excessive ove:r- shoot caused by direct implementation of the integral action. The amount of maximal saturation of the inte- gral action is studied by using the squeezing technique approach. The stability analysis of the PI/PD part is also discussed.

2

logic systems Topology of the L,

neural networks and fuzzy

2.7 The L,neural networks (L, NN)

A great number of multilayered feedfonvard neural networks (MLNNs) have been widely discussed in recent literature [13, 141. An MLNN includes an inpiut layer, an output layer and a number of hidden layers. Each input layer is composed of input nodes. Each hid- den layer (and output layer) consists of processing units (the so-called hidden nodcs or basis functions), and each unit is the composite of an activation function (AF) and a transfer function (TF).

The input signals to each processing unit are first transformed into an activation level via the activation function. The activation level is further mapped into a crisp output by the transfer function. The aim of those mappings is to store the input-output relationship in parameters of the AF/TF via learning. With various selections of activation functions and transfer func- tions, a variety of MLNNs could be synthesised.

Without loss of generality, the activation function V(x; p) could be generally expressed in quadratic form:

( I ) where denotes the vectodmatrix transpose, x

2

[x,, ..., xIlT E R‘ is the input vector, c

g

[c1, ..., c1lT E R’ is

called the centre, A is an I x I semi-positive-definite matrix; ?L [ I , , ..., hIlT E R‘and p = overall parameters

of an AF. Some popularly used AFs are special condi- tions of such a quadratic form. For example [13-151,

a U ( X ; p) = (X - c ) ~ A ( x - C)

+

XTx

+

b

4 x ;

PI

[k

Xixi

+

6 i f A = O i=l (linear) C ( X Z - if X = 0, 6 = 0, A = diag - [ % = I S2 (spherical)

LI

As to transfer function @(.), all TFs mentioned in the literature are analytical almost everywhere [16-181. Far instance, (linear)

f ”

1 1

+

exp(-v) (sigmoidal) 4 ( U ) =

I

(3) n

I

e x p ( j u ) , j = (Fourier)

I

exp(-u) (inverse exponential) where e x p b ) might include cos(v), sin(v) and their lini- ear combinations. Combining any of the AFs and any IEE Proc -Control Theory Appl., Vol. 143, No. 2, March 1996

of the various possible TFs, a processing unit with var- ious characteristics could be synthesised. The typical processing unit of a BPNN, for instance, is the combi- nation of a linear AF and a sigmoidal TF, and the typ- ical processing unit of an RBFN is the combination of a spherical A F and any analytical TF.

Each layer of a multi-inputhingle-output (MISO) MLNN is composed of various processing units with a suitable choice of AFs and TFs. Without loss of gener- ality, the three-layered MISO MLNN with I inputs and J hidden processing units would be used in the follow- ing, and both the AF and the T F in the output layer are assumed to be linear throughout this work. Each processing unit combining an AF and a T F in the hid- den layer is called the basis function @j

e

@(v(x; pJ), where pi denotes overall parameters of the jth process- ing unit in the hidden layer. Such a neural network is mathematically equivalent to a finite-dimensional func- tion space spanned by neural basis functions:

J

f ( x ;

P)

2

wjd(”(x; Pj)) (4)

j=1

where P denotes overall network parameters.

The bounded input-bounded output (L,) RBFNs and BPNNs, i.e. L, NNs, would be used in this paper. Without loss of generality, the basis function Qj for an L, NN is assumed to be normalised for allj, i.e.

ll$jjll,

= 1 for allj. The relationship between the norm of the weight vector and the range space of an L, N N could be derived from the Minkowski inequality and func- tional analysis. Let Q2, ..., QJ} be a basis for an L, NN. Then for every choice of weights wl, w2, ..., w J one has

J J

aCI1LA’

I

l l ~ ~ ~ ~ ~ ~ l l c c

5

CI%

(5)

j=1 j=1

where 0 < a 5 1. The inequality, in fact, could be viewed as the definition of an L, NN.

2.2 The fuzzy logic systems (FLS)

A typical FLS is composed of four principal compo- nents [11]: (1) a fuzzifier; (2) a rule base; (3) an infer- ence engine; (4) a defuzzifier. The fuzzifier deals with mapping scaled input variables and transforming the mapping into appropriate linguistic values. The rule base comprises the well established knowledge of the application domain. For an inference engine, it emu- lates human decision-making logic. Such operations are performed by employing fuzzy implication and fuzzy logic inference. As the final stage of the FLS, the defuzzifier generates a single crisp output from the inferred fuzzy action.

Without loss of generality, an FLS with the singleton fuzzifier, Gaussian membership functions, the product inference rule, the centre average defuzzifier and the equal area of membership functions for all output lin- guistic terms would be used in this paper. The output of such an FLS with J inference rules is [l 11

&;P) =

J

j=1

where wj is the corresponding crisp output of @(v(x; pj)) with unit membership function value. P is the set of all adjustable parameters in the FLS. The FLS could be viewed as a modified Gaussian potential function net- work (modified GPFN [13]). Each rule in the FLS is functionally equivalent to one hidden node of a modi- fied GPFN. The wjs here refer to the weights of the

FLS in accordance with neural network jargon. The discussed FLSs are notably input-output bounded, and have the relationships with output weights as demon- strated in eqn. 5. (Ia(t)

w

... ~ ... ~ ... VSPID . .- . . . . Fig. 1 VSPID control system

01 y ' ' ' I ' 1 I I I

0 1 2 3 L 5 6 7 8 9 10 Characteristics of a(t) with E = 1.0 and various 77 values

le(t)l Fig.2 -q = 0.5 -q = 1.0 I q = 2.0 3 (neural/fuzzy VSPID)

3.

I

The variable structural PlD controller

(VSPID)

The system has notably no steady-state error in response to a step input disturbance provided that the control law employs the integral mode. Undesirable overshoot, increasing sharply as a function of the gain, will occur for sufficiently large integral action. Sharply increasing the controller gain could significantly accel- erate the system response in the absence of the integral mode, but a steady-state offset would be displayed in such a case; thus, reaching a compromise is vital. A nonlinear variable structural PID controller could be used for such a correcting effort, as is depicted in Fig. 1:

u ( t ) = E + k c ( t ) e ( t )

+

a k d ( t ) r ( t )

+

(1 - a ) k i ( t ) m ~ ( t )

Neural/fuzzy variable structural PID controller

= G

+

a [ k , ( t ) e ( t )

+

k d ( t ) r ( t ) ]

+

(1 - a ) [ k , ( t ) e ( t )

+

7ci(t)ml(t)] _{( 7 )}

202

where U is a constant, r ( t )

2 de(t)/dt, mkt) &

e(z)dz and a E [0, I]. The controller would turn out to be either a PD or a PI controller if a is either 1 or 0.

Instead of a drastic change of a value, a could be rea- sonably defined as

4 4

= tanh(rP(t)) (8)

where

a is an increasing function of le(t)l, and converges to either 1 or 0 if le(t)l approaches infinity or le(t)/ enters the tube 0 I le(t)l 5 E such as shown in Fig. 2, i.e.

1 if le(t)l

>>

E

0 if /e(t)l

5

E

a ( t ) =

The q value in eqn. 8 determines how quickly a

changes between zero and one. For reasonablc q val- ues, the VSPID would behave from the PD controller in the case of large error to the PID case and the PI case, i.e.

(

k c ( t ) e ( t )

+

k d ( t ) T ( t ) for le(t)l >> E

An infinite q value would lead the VSPID to be either a PD or a PI controller according to the magnitude of le(t)l. The time at which the controller structure change occurs is thereby determined by a flexible program which is directed by the current value of the error sig- nal.

In order to put into operation the proposed VSPID controller in the practical process, stability analysis in the PD control mode and training procedures of the proposed controller parameters are discussed in the dis- crete time domain. The parameters of the proposed VSPID controller could be updated for every control interval T. A zero-order holder is used to keep a con- stant controller output during each interval. The nota- tions therein are defined as follows:

4 n ) = { s p ( t ) - v(t)}lt=nT e ( n ) - e ( n - I) T r ( n ) = n k=l u ( n ) =

a

+

k ( n ) e ( n )

+

a(n)r(n)

+

(1 - a ( n ) ) r n ~ ( n )

sp(t) and y ( t ) are reference and process output; e(n), r(n), min) and u(n) are error, rate of change in error, integral of error and controller output, respectively, at the nth sampling point.

3.2 L, NN/FLS based variable structural PID controllers (L, NN- VSPID, FL- VSPID)

The neuraufuzzy VSPID has a structure similar to that of the conventional nonlinear VSPID controller, but its parameters are changed according to local conditions (see Fig. 3). Here,

L,

NNs or the previously mentioned fuzzy systems are used for constructing a nonlinear relationship between controller parameters and local control conditions. Parameters of the

L,

NN-VSPID

or the FLVSPID are defined as

k ( t )

= f ( x ; P1)

k ( t )

= f ( x ; P2) (13) IC&) = f ( x ; P3) where f ( x ; Prn)

Ewz3#(v(x(t); pz3)) for the L , NN-VSPII)

zE17,3EJm

x w z g 4 ( M t ) ; P z 3 ) ) (14) for the FL-VSPID, z E L , JEJ,,,

E

E 4 ( ~ ( x ( t ) ; P z 3 ) ) = 1 , 2 , 3 % € I , JEJ,

Hence, k,(t), k,(t) and k d t ) are outputs of three three- layered L, NNs or three FLSs with two inputs x(t) = [e(t), r(t)lT. p2] denotes overall parameters in the ijth hidden processing unit or the ijth rule.

1

\

neurallfuzzy VSPID

Fig. 3 Neural/fuzzy VSPID control system

For the L, NNVSPID/FL-VSPID, v(x(t); pii> is the activation level of the ijth hidden processing unit

M t ) ; Pzj) (1.5)

-

-

[

(vs+

f y r

for the L , RBFN/FLS

A & e ( t )

+

X r j r ( t )

+

bi, for the BPNN

Cp is the analytical function for the L, RBFN, and Cp is the sigmoidal function for the BPNN. As for the FLS, Cp is the discussed fuzzy basis function. pij = [cei, c . sei,

the BPNN.

A$

or

1;

refers to the weight in the hidden layer that connects the ijth hidden node to the input node of e(t) or r(t). Subscript ei denotes the ith node in the e direction, and subscript rj denotes the jth node in the r direction for i E 11, 12, I3 a n d j E J1, J2, J3, where

11

b

(1,

...,

111, Ji { I , ..., &2}, 12

b

(1, ...,

41,

52

k

(1,

..., 14}, I3 { 1, ...,

Z5}

and J3

k

{ 1,

...,

16}. The lis refer to the number of AFs in either the e ( i = 1, 3, 5 ) direc- tion or the r (i = 2, 4, 6) direction. Tuning parameters

of the specific L, NNs/FLSs in k,(t), ki(t) and kd(:t)

might encompass centres [ceb cTjlT, widths of receptive field [sei, srjIT, connective weights in the hidden layer srjIT for the L, RBFN/FLS, and pij =

[Ai: Ai;,

bij]

!F

fix

IEE Proc-Control Theory Appl., Vol. 143, No. 2, March 1996

[ h i ] and

[Ai;],

bias terms [by] and the output connective weights [wV] for i E 11, 12, I3 a n d j E J1, J2, J3.

Outputs of controller parameters and output connec- tive weights of associated NNs/FLSs have the following relationships:

'2 Iwz31

5

I l k t ( t ) l l a

5

I w Z J (16)

zEIz 3 E J z ~ € 1 2 3 E J z

a3 l W v I

I

Ilki(t)llm

I

t:

I w z g I

% € I 3 3EJ3 ~ € 1 3 3 E J 3

where 0 < a l , u2, u3 I 1. Those relationships imply that parameters of the L, NN-VSPID/FL-VSPID control- ler are bounded.

An L, RBFN-VSPID or a FLVSPID is function-

ally equivalent to a linear VSPID if all receptive field widths approach infinity in neural/fuzzy basis func- tions:

Property 1 (L, RBFN-VSPID and FL-VSPID are functionally equivalent to linear VSPID): An L, RBF- N V S P I D or an F L V S P I D controller is functionally equivalent to a linear VSPID controller if sei, srI

-+

M b' i E 11, 12, I,, and b ' j E J,, J,, J3.

Proo$ Each basis function in an L, RBFN or an FLS becomes a constant function, providing that all recep- tive field widths in each basis function approach infin- ity. Hence, an L, RBFN-VSPID or an FL-VSPID is functionally equivalent to a linear VSPID controller if se,, srl

+

00 b' i E 11, 12, I3 and ' d j E J1, J2, J3.

Property 1 gives the way to set L, RBFN-VSPIDI F L V S P I D parameters initially by letting the receptive field widths in the neuralifuzzy basis function be large enough so that the initial performance of the L, RBF- N-VSPID/FLVSPID could be like the performance provided by the linear VSPID.

j neurallfuzzy VSPID j

Mod$ed neurawuzzy VSPID control system Fig. 4

4

Integral or reset action is usually taken to eliminate the steady-state error in feedback controllers. One of the penalties that must be paid for this convenience is 'reset windup' or excessive overshoot caused by the direct Nonlinear PI controller with antireset windup

implementation of the integral action. Installing a lim- iter on the controller output to keep it from going beyond the operating range of the actuator is a way to prevent reset windup. To do this, let us first break down the nonlinear PI control rule into its parts (i.e. see Fig. 4).

Suppose a saturating actuator with a positive upper bound Mu and a lower bound M' is installed. Then the output of the saturating actuator is

M u for m(t) 2 M u u s ( t ) = m ( t ) for M'

5

m ( t )

5

M u (17) M' for m(t)

5

M' (18) 7jZ1(t) = e ( t ) (19)

i

where m ( t ) = U + k c ( t ) e ( t )

+

k i ( t ) m I ( t )

and W 2 U 2 M1 2 0. From the definition of mXt) we get

Since kc(t) > 0 and k,(t) > 0 in nature for all t, the reset-feedback implementation of the time-varying PI control law is thus given by

1

7 j Z I ( t ) = - ( m ( t ) - ki(t>rnI(t) - U ) (20)

k C ( t )

Solving the above time-varying first-order ordinary dif- ferential equation we have

Since -(U - M') 2 m(t) - U 4 Mu - U for all t , thereby

exp

(-

Z d r ) -(U - M l ) k," Then Thus Therefore (U -

W ) -

k y k ;

Describing the exact trajectory of mxt) needs full information of kc(t), k,(t) and m(t). Estimating the dynamics of kc(t) and k,(t) is difficult if kc(t) and k i t ) are not prescribed. The possible extreme output at some instant and steady-state behaviour of m i t ) is investigated in the following.

There exist values of k / , k;, k,' and k," for the pro- posed neural/fuzzy PI controller such that

0

<

k: 5 k Z ( t ) 5 ky and (22) 0

<

rk: 5 rkc(t) 5 ik; Then we have k: k i ( t )

<

2

o < - < -

k," - k c ( t ) - k:

' i t

(23)

For all t and s I t, it thus implies that

Hence 0

<

exp

(-

1

Z d r ) exp

(-

[

Z d r ) O < Then

5

k;(t)rn~(t) 5 ( M u - U ) k (1 - exp ( - $ t ) ) (31)

where k f k;"k,"lk,'k,' and k > 1. The result is consist- ent with the initial condition ki(0)ml(O) = 0. The deriva- tion thus implies

To this end, Property 2 is thus derived.

Property 2 (Integral action of the neuralfuzzy VSPID

controller is bounded): Since there exist constants k / , k,", k,' and k," for neurallfuzzy VSPID controller parameters such that 0 < k," I k,(t) I k," and 0 < k,' I kc(t) I k," for all t , with the proposed antireset windup implementation the integral action of the neural/fuzzy VSPID controller is bounded.

The possible maximal output of k,(t)ml(t) is therefore equal to k ( W - U), and the possible minimal output is -(U - M')/k. The rate at which the controller output,

saturated at Mu, is reset is governed by the gain k , which determines how quickly the integral is reset, i.e. for large k it takes much time to reset the integral action and vice versa. Likewise, the rate at which the controller output, saturated at M1, is reset is decided by Ilk, which determines how quickly the integrator is reset, i.e. less time is spent to reset the integral action if Ilk is close to one and vice versa. Thus there is no effect on normal operation when the actuator does not saturate. Whilst saturating occurs, the feedback signal will try to drive the integrator to a value such that the controller output is exactly at the saturation limit. Preventing the integrator from winding up is therefore IEE Proc -Control Theory A p p l , Vol 143, No 2, March 1996 204

clear. There would be no saturation effect if

k

were equal to one. The result is consistent with the fact derived from linear control theory.

At steady state the error signal must be zero, since e(t) = mXt) = 0. No steady-state offset is therefore guaranteed for this modified PI control law. Suppose k,(t) = k," and k,(t) = kj" at e = r = 0. It implies that

M L

5

m(o0) = k,-rnl(oo) + U

5

M U (33) The integral signal is confined to the range of the satu- rating actuator at steady state.

With the antireset windup, the nonlinear PI contrd- ler is always input-output stable, i.e. for finite energy input, the PI controller with antireset windup would yield finite energy output. A wide enough operating range of the controller is therefore required to put the controller into operation.

Suppose the nonlinear PI controller is operated at the normal level. Then for a given process, linear or non- linear, the proposed nonlinear PI control system hias the same local stability (asymptotically stable or unsita- ble) at the equilibrium point, sp, as the linear PI (k, = k,", k, =

e)

control system does. This is because the linearisation of the proposed nonlinear PI controller around the equilibrium point results in the linear PI controller. The two controllers thereby behave similairly in the region around the equilibrium point. According to Lyapunov's indirect method [19] (Lyapunov's first method on stability), the neural/fuzzy PI mode is asymptotically stable (unstable) at the equilibrium point if and only if the linear PI control system (k, = k,", k , = k;) is asymptotically stable (unstable).

5

control law

Stability analysis of the neural/fuzzy PD

The closed-loop system is L stable if both subsystems, the process and the controfer, are Lp stable in thein- selves and if the 'loop gain' is less than one for 1 I 1;' I

M [19]. Given any process in a complete function space,

one could thus find a controller in the complete space such that the composite of the process and the control- ler converges at a fixed point in space, providing the gain of the composite is less than one. Suppose the controlled process is L, stable. The proposed neural/ fuzzy PD control law would be one of the choices since the PD controller is L, stable, an assertion which is discussed in the following.

Consider the neural/fuzzy PD controller output at the nth sampling point:

u ( n ) = U

+

k,(n)e(n)

+

kd(n)~(n) Then for n 2 1 we have

lu(n)l

I

Ial

+

Ikc(n)Ile(n)l

+

Ikd(n)ll+)l

Therefore for n 2 1 S U P

I4n)

I

IEE Proc.-Control Theory Appl., Vol. 143, No. 2, March 1996

+

ZE13jEJ3

T sup le(. - I ) /

+

By using the definition of the L, gain of an L,map- ping

Ax(

t)) defined in [ 191,

where y, and b, are finite constants, the derivation leads to Property 3.

Property 3 (Gain of the neuralfuzzy PD controller): The gain of the PD controller yJu) is less than or equal to

Ym(f) = inf{ym : SUP

If1

I

Ym SUP

14

+

bm} (37)

Since there exists a finite upper bound of the pro- posed neural/fuzzy PD controller gain, an application of the small gain theorem yields the following stabilisa- tion criterion for the proposed neural/fuzzy PD control system.

Property 4 (Suflicient criterion for the stable neural/ fuzzy PD control system) : The proposed neurallfuzzy

PD control system is stable if

c

i E I l

where y,b) is the process gain.

Centres and receptive field widths in the neural/fuzzy basis functions are irrelevant to the stabilisation crite- rion. The most important factors related to the stabili- sation criterion are output connective weights of the neural/fuzzy systems. According to Property 4, the number of parametric space dimensions related to the stability region of the proposed PD control system is reduced to I I x l2

+

Is

x 16, as compared to the dimen- sions ( I I x l2

+

211

+

212

+

Is

x l6

+

21,

+

2 4 ) in the orig- inal neural/fuzzy parametric space (PD part). Stabilising the PD mode control could be done by appropriately adjusting weights in k , and kd satisfying the stabilisation criterion.

6 Training procedures

The aim of training a neural/fuzzy VSPID controller is to minimise the following performance measure, i.e. to solve a finite horizon optimisation problem

p N P

p = l n=l

( 3 8 ) T

Je =

2

c

{[sdn) - Y(n)I2),

Here, {sp(.>}, E S, S is a set of P individual reference inputs to the process over the possible operating region; {U(.)}, are outputs of the real plant in response to the given pth testing input excitation; Np is the total number of control instants within the interval of each training time under consideration. Different choice of the input set for training could lead to somewhat dif- ferent controller parameters. One can directly use any existing optimisation methods, such as the steepest

gradient method or the conjugated gradient method, in search of the optimal parametrisation of the neural/ fuzzy VSPID controller.

Suppose the controlled process has a time delay d. Let zi be one of the elements in parametric space of the proposed neural/fuzzy VSPID controller. The associ- ated gradient is

p NP

3

= -T

{

[ ( s y ( n

+

d ) (39)

p = l n = l

az,

In the equation, dy(n

+

d>/du(n) required to update parameter zi in the parametric space of the neural/fuzzy VSPID controller during optimisation. Therefore, a first-principle mathematical model [20] or a well trained plant emulator [21] is required to estimate these partial derivatives of the plant response to neural/fuzzy VSPID controller parameters at current operating points. However, especially where more complex plants are concerned, such a precise plant emulator is often not available. When the information is not provided, one can take the sign of the partial derivatives of the plant, i.e. sign dy(n + d)/du(n), to give approximate informa- tion [22]. Such an approximation might, however, slow down the speed of convergence in finding optimal solu- tions.

Several training passes are required to search for convergent parameters (q*, E*, ce:, er;*,

se,,

sr; and w;,

for i E 11, 12, 13, j E J1, J2, J3) of the optimal neural/ fuzzy VSPID controller (U*). Notice that the current VSPID controller parameters are kept constant through the whole control pass in the course of this searching. Given the specific input pattern x(t) = [e(t), r(t)lT, the control response of the optimal neuraufuzzy VSPID controller is

U* (n) = U

+

_(n)e(n)

+

_{a*(n)k;(n)r(n)}

+

(1 - a!*(n))k:(n)rnl(n) (40)

7 Illustrations

7,

I

pN control of a neutralisation process

The applicability of the neuralifuzzy VSPID controller is demonstrated by controlling the pH value of a neu- tralisation process. The control near the neutrality point (pH = 6

-

S) is notably difficult since a slight change of operating condition could make a drastic dif- ference in pH value.

F= Fit F;! pH=-log [ H’I

Fig.5 p H control system

206

In this example, the strong base NaOH is used in controlling the acidity of a wild process stream with weak acid HA. A perfectly effective liquid level con- troller is used to keep the reactor volume constant (see Fig. 5). The balanced equations are [23]:

Acid balance:

Sodium balance:

Charge balance:

[H+I +

P+l

{

C + K )

+

[H+] {

(C

- <) K , - Kw } -K, Kw z 0

(43) Sensory signal lag:

p H ( t ) = - loglo([H+](t - 0.4)) ₍₄₄₎

In these equations,

5

_{[Nuc], Ku} =

[H+][A-]/ [HA] (the acid equilibrium constant) and K, = [H+][OH-] (the water equilibrium constant). The charge balance equation results from the fact that [

+

[H+] = [OH-]

+

[A-]. The operating range of the base inlet stream F2 is 0

-

2L/min. Numerical values of the phys- ical variables involved here are listed in Table 1.

[HA]

+

[A-],

5

Table 1: Nominal values of parameters in the neutralisa- tion process K, = IO”mol/L K,,, = 10-14m012/L2 Cl = 0.10998moI/L C, = Imol/L pH = 4 F2 = 0.9988Umin 5 = O.Imol/L

5

= 0.09081mol/L Fl = 10L/min V = 50L

In demonstrating the applicability of the proposed control method, the set-point of pH is changed from 4

(initial steady state) to 7 and two input loads, acid con- centration from 0.10998 to 0.1 105mol/L and acid flow rate from 10 to 10.5L/min are then introduced at 60 min and loomin, respectively. The control interval T i s 0.2min. 10 9

Z 8

7 6 5 4 0 20 40 60 80 100 120 140 time, min

Fig. 6 Compar~on of servo and regulatory control performance for the n e u t r a l i s a t i o n 7 when using optimised linear PI and optimised L, NN-PI: p against time

linear PI -

L, NN-PI -

Two 5 x 5 (i.e.

Zi

= 5, i = 1, ..., 4) modified GPFNs are used to construct a nonlinear relationship between local control conditions and L, NN-PI controller parameters. Optimisation is achieved for the linear PI and L, “PI by using the steepest-descent technique with the criterion of minimum integral of square error. Figs. 6 and 7 show servo and regulatory control results IEE Proc.-Control Theory Appl., Vol 143, No. 2, Murch 1996

derived from using both the optimised linear PI con- troller and the optimised L, NN-PI. The k, and ki tra- jectories of the optimised L, NN-PI are illustrated in

Fig. 8. 0.0060 0.0055 0.0050 0.0045 0.0040 0.0035- 0.0030 0.0025- 0.0020 0.0015- 0.0010 1.25 I I I I I 1 I I 1 I I I I I - - -

-

7

-

c

A. -

izz

hnnn*l---- I I I I 0.95 l . O 0 L - - - J 0 20 40 60 80 100 120 140 time,min

Fig. 7 , Comparison of servo and regulatory control performance for the

neutrahsation process when using optimised linear PI and optimised L, NN-PL F2 against time linear PI - L, "-PI - 0.0050 I I I I I I I 0.0045

-

h I I I I I 0 20 40 60 80 100 120 140 time, min 0.0010 Fig.8 k o f L N N P I - k; of L z "PI: -

k, and k, trajectories of optimised L, NN-PI

Three 5 x 5 (i.e.

Zi

= 5, i = 1, ..., 6) modified GPFNs are used to construct a nonlinear relationship between local control conditions and L, NN-VSPID controlller parameters. Fig. 9 shows servo and regulatory contr'ol results derived from using both the optimised linear PID controller and the optimised L, NN-VSPID. The k,, k, and kd trajectories of the optimised I,, "_VSPID are illustrated in Fig. 10. The variation of the a value for the optimised L, NN-VSPID is illus- trated in Fig. 1 I. The best obtainable servo and reguLa- tory control performances for the optimised linear PI, the linear PID, the L, NN-PI and the L, NN-VSPID, are shown in Table 2.

Undesirable overshoots and large transients, iIS

observed from the control performance derived from conventional controllers, can be satisfactorily elimd- nated by the neural/fuzzy VSPID controller. The improvement in servo and regulatory control perfornn- ance derived from the preassigned algorithm of the law of the controller structure change and the resilient re1,a- tionship between controller parameters and local coin- trol conditions is desirable and excellent. Satisfactoiry servo and regulatory control performance could also be

derived from FLVSPID controllers. This example has illustrated the potential value of using the proposed L, NN-VSPID controller in highly nonlinear chemical processes.

time, min

Fig.9 Comparison of servo and regulatory control performance for the neutralisation process when using optimised linear PID and optimised L, NN-VSPID

linear PID -

L.=.= NN-VSPID -

Fig. 10

k, of L, NN-VSPID ~

kc, K, and kd trajectories of optimised L, NN-VSPID

k; of L.. NN VSPID - k> of NK-VSPID

-

a 0.4 0.21 I1 1 I I I I I 1 I 0 20 40 60 80 100 120 140 160 t i me, m in

Fig. 11 Variation of a value for optimised L, NN-VSPID

8 Conclusion

This paper has described the design method of a neu- ral/fuzzy variable structural proportional-integral-deriv- ative (neural/fuzzy VSPID) control system. The PD mode is emphasised in the case of a large error occur- ring so as to speed up response, and the PI mode is applied to small error conditions to eliminate the steady-state offset. A sigmoidal-like neuron has been employed as a preassigned algorithm of the law of the structural change. Meanwhile, the VSPID controller

Table 2: Comparison of servo and regulatory control performance for the neutralisation process when using the linear PI, the linear PID, the! L, NN-PI and the L, NN-VSPlD

Controller (optimised) k, k, kd E Je

linear PI 2.673 x 10-3 I .9:38 x 10-3 - - - 294.31

linear PID 2.436 x 10-3 2.265 x 10" 5.399x IO" - - 270.82

L, "-PI trajectories of k, and k, : see Fig. 8 - - - 129.45

L, NN-VSPID trajectories of k,, ki, kd and a : see Figs. IO, 1 1 10 2.951 106.21

parameters are changed according to local conditions. L, neural networks or bounded-inputhounded-output fuzzy logic systems have been used for constructing a nonlinear relationship between the PID controller parameters and local operating control conditions. The

PI mode with antireset windup has been designed for prevention of excessive overshoot caused by direct implementation of integral action. The stability analysis of the PI/PD part has also been discussed. A neutrali- sation process has been used to demonstrate the appli- cability of such a controller for controlling highly nonlinear processes. 9 1 2 3 4 5 6 7 8 9 ferences

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