Nonlinear internal model control using local model networks

Nonlinear internal model control using local model networks M.D. Brown G.Lightbody G.W.Irwin

Indexing terms: Internal model control, Neural networks, Nonlinear control, Nonlinear modelling

Abstract: Local model (LM) networks represent a nonlinear dynamical system by a set of locally valid submodels across the operating range. Training such feedforward structures involves the combined estimation of the submodel parameters and those of the interpolation or validity functions. The paper describes a new hybrid learning approach for LM networks comprising ARX local models and normalised gaussian basis functions. Singular value decomposition (SVD) is used to identify the local linear models in conjunction with quasiNewton optimisation for determining the centres and widths of the interpolation functions. A new nonlinear internal model control (IMC) scheme is proposed, based on this LM network model of the nonlinear plant, which has the important property that the controller can be derived analytically. Accuracy and stability issues for the nonlinear feedback control scheme are discussed. Simulation studies of a pH neutralisation process confirm the excellent nonlinear modelling properties of LM networks and illustrate the potential for setpoint tracking and disturbance rejection within an IMC framework.

1

Introduction

In an attempt to accurately model nonlinear systems, a wide variety of techniques have been developed, such as the Volterra and Wiener series [l] or NARMAX modelling [2]. Such approaches have had limited success in industry, owing primarily to their complexity [3]. Recently, neural networks have generated considerable interest as an alternative nonlinear modelling tool [4]. Utilising the ability of the neural network to approximate arbitrarily nonlinear vector functions and combining this with dynamic elements such as integrators, filters or delays, yields a powerful, yet easily applied modelling technique [5, 61. Bhat et al. [7] have successfully applied the multilayer perceptron network 0IEE, 1997 IEE Proceedings online no. 19971541 Paper first received 27th January and in revised form 10th July 1997 The authors are with the Advanced Control Engineering Research Group, The Queen’s University of Belfast, UK M.D. Brown is now with the Department of Mechanical Engineering, University of Leeds, UK IEE Proc.-Control Theory Appl., Vol. 144, No. 6, November 1997

(MLP) for the modelling of several chemical processes. Likewise Lant et 01. [8] have shown that neural network modelling methods can be used to provide estimates of primary variables from secondary data for a biomass application. Willis et al. [9] and Turner et al. [lo] have shown the potential of dynamic networks for pilot and industrial distillation column modelling, respectively. Qin and McAvoy [l 11 developed methods for neural modelling from correlated plant data with application to a number of MIMO industrial processes. An excellent understanding of the benefits and drawbacks of neural modelling based on simulated examples and an industrial distillation column is presented in Pollard et al. [12]. Neural networks are now also being used in modelbased control schemes. These techniques traditionally utilise some parameterised model of the plant from which a suitable control law can then be designed. It is common that both the model and controller are linear, with the parameters of each adapted over time to account for plant nonlinearities. Narendra and Parthasaranthy [5] proposed the enhancement of model-based control strategies such as indirect adaptive control, by replacing the linear models with nonlinear neural network representations. In a similar manner, Montague et al. [13] have extended the generalised predictive control scheme, and Hernandez and Arkun [14] have extended dynamic matrix control by using a nonlinear predictive model of the process based on feedforward neural networks. One particular control approach for which a well understood framework exists for the structured utilisation of nonlinear plant models, is internal model control [15]. It is within this control framework that the benefits of nonlinear neural modelling can most readily be applied, particularly in application to the chemical process industry [161. The application of neural networks for modelling and control, however, does have some fundamental limitations. First, the nontransparent, black-box approach makes it difficult to incorporate a priori system information, and to interpret the final structure in terms of the physical characteristics of the process under consideration. Secondly, neural network modelling fails to exploit the significant theoretical results available in the conventional modelling and control domain, making it very difficult to analyse their behaviour and to prove stability. Also, since most industrial processes operate under feedback control within a small region about any given operating point, the online training of neural networks must be approached with care. This is especially true for networks with global support functions such as the MLP, since behav505

iour learned in one region will degrade when the operating region changes. An alternative approach that facilitates online training is to use neuron support functions that are active only over a small or local region of the network’s input region. Networks of this type include the gaussian RBF (radial basis function) [171, CMAC (cerebellar modular articulation controller) [181 and B-(basis) spline networks [19]. The advantage of these local structures is that training involves linear optimisation (unlike the MLP) and can be performed online [20]. However, the major drawback of such networks is that a restrictive number of processing units may be needed to accurately represent the nonlinear dynamics over the full operating region. Recent research has suggested that fine partitioning of the input space, in which each neural basis function is active, is not necessary. Rather, it is sufficient that the partitioning procedure should simply split the input space into the expected operating regions of the plant. The local model network adopts this philosophy by forming a global system model from a set of locally valid system submodels [21]. The general feedforward neural type structure of the local model network consists of submodels that could themselves be neural networks such as the MLP or RBF. Indeed, linear models and a priori system information from a physical modelling exercise could be incorporated within this structure. The outputs of each submodel are passed through a local processing function that effectively acts to generate a window of validity for the model in question. These nonlinear weighting functions utilise only a subset of the modelling data available to generate the desired partitioning of the model space. The resultant localised outputs are then combined as a weighted sum at the model output node. As with conventional neural networks, training is a crucial issue for LM networks since there is the added complexity of identifying the local models as well as the parameters of the interpolation functions. This paper describes a new hybrid learning approach for LM networks constructed from ARX local models and normalised gaussian basis functions. Singular value decomposition is used to identify the local linear models in conjunction with quasiNewton optimisation for determining the centres and widths of the interpolation functions. To avoid overtraining problems in nonlinear dynamic modelling with noisy data, the SVD minimises a one-step-ahead prediction error, while the nonlinear optimisation is performed on a model based error. 2

Local model networks

The representational ability of radial basis function neural networks can be generalised to form the local model network structure. The basis functions in this instance are used to weight general functions of the inputs as opposed to simply the weights. The network output can be described as [21] M

b = F ( @ 4, ) =

c

fz($)P2(4))

(1)

2=1

where the M local models J;(y)are linear or nonlinear functions of the measurement vector y, and are multiplied by a basis function p,(@)that is a function of the current operating region vector $, This operating region does not necessarily need to be the full model 506

input vector y but can be a subset of the measurement data available. In the RBF neural network the functions f,(y)are constants and the basis functions pi(@) are radial. In the LM network the basis functions pi(@) are commonly chosen to be normalised gaussian functions, i.e.

c

exp(-ll$ - c 2 1 1 2 / 2 d

1=1

Thus pz(@) can be interpreted as a function that gives a value close to 1 in parts of @ where the local J1 is a good approximation to F, and a value of 0 elsewhere. If, for example, the local models are of the ARX form given in eqn. 3 ,

f%(@) = bozu(k)+ b l , U ( k +Ul,Y(k)

- 1)

+ . . . + b,,u(k

-

s)

+ a22?4(k- 1)+ . . . + qT+1)%Y(k ). -

(3) where r and s are the orders of the plant output and input y ( k ) and U@), respectively, at time k , then the LM network represents a nonlinear ARX model as follows:

ji(k + 1) = F ( $ ( k ) ,$ ( k ) ) @(W = [Y(k),Y(k - 11,.. . , Y &

-.I,

u ( k ) ,u ( k - l ) ,. . . , u ( k - s)]T

4(k) c $(k) (4) Combining eqns. 1 and 4 yields the following expansion: $(k

+ 1) = Bou(k)+ B1u(k - 1) +. . . + B,u(k s) +AlY(k) + A2Y ( k 1) + . . + A(T+l)Y(k - ). -

-

(5) The resulting model is still ARX in structure, with the parameters A, and B, determined at each operating point @(k)as follows: M j=1

j=1

The underlying assumption for the local model network strategy is that the systems to be modelled undergo significant changes in operating conditions. For most batch and continuous processes in the chemical, biotechnological and power industries, definite regimes can be identified during procedures such as start-up, shut-down and product shifts. The complexity of the overall model can be reduced by incorporating simpler models in each operating region. For example, local state space and ARMAX models can be formed using localised perturbation signals and then interpolated to give global nonlinear state space and NARMAX (nonlinear ARMAX) models [22]. The identification of local operating regimes for an unknown plant is difficult. Any such identification strategy has to take into account the complexity of the target mapping, the representational ability of the local models associated with the basis functions and the availability of the data. The problem is therefore to identify those variables which describe the system operIEE Proc-Control Theory Appl., Vol. 144, No. 6, November 1997

ating behaviour. A priori knowledge of the plant can be used at this stage. When little knowledge of the actual regimes exists, however, it may be beneficial to use unsupervised learning methods, such as k-means clustering and nearest neighbours, to give an initial estimate of the normalised gaussian activation regions. These clustering methods are valid in this case since many plants tend to operate in distinct regions. Despite these difficulties, successful applications of the local model technique have been reported in the biotechnology and chemical engineering industries [23, 241. 3

Hybrid learning scheme

3.1 Optimisation algorithm

Referring to eqn. 1, if we define _a = [c1, c2, ..., cM, q, 02,..., oMIT as the centres and widths of the normalised gaussian interpolation functions, a cost function can be defined as N

/ / M

i=l

\ \j=1 (7)

where N is the number of training vectors and y i is the scalar plant output, and hence the target output associated with the ith training vector. A hybrid optimisation strategy similar to that found in [25] for minimising eqn. 7 can be derived by considering a nonlinear technique that finds the centres and widths of the normalised gaussian interpolation functions, and a subsequent linear optimisation of the local linear models at each iteration. Structuring the overall optimisation in this way can lead to a number of advantages in training the network. First, the number of independent parameters in the network is reduced. It should be expected therefore that the time taken to reach a minimum of the nonlinear cost function will be reduced. Secondly, the network is always in a state where the error is at a global minimum in the space of the local models, since these are obtained by linear optimisation. The hybrid optimisation strategy proposed in this paper uses a secondorder gradient descent method for determining the nonlinear parameters, with singular value decomposition used at each iteration to determine the linear models, as follows. The gradient with respect to _a of the cost function defined by eqn. 7 is given as

Partial differentiation with respect to a! of the ith local model network output (with normalised gaussian activation functions and local linear models) is given by the expressions

IEE Proc.-Control Theory Appl., Vol. 144, No. 6, November 1997

i = ( 1 7 2 7 . . ,ly7 IC = ( 1 7 2 7.., . K) m= (l,27...,M) (9) where N is the number of training vectors, K is the dimension of the operating regime, and A4 is the number of local models. These equations are evaluated over the N training patterns and combined to form the batch gradient given by eqn. 8. Having determined the cost function and its gradient, any suitable gradient descent algorithm should succeed in reducing the cost function to at least a local minimum. This study employs a second-order gradient technique that is much more powerful than simple gradient descent or error backpropagation. The parameter update equation is given by

rllcH,-’(a)VJ,,(a) (10) H i s the Hessian matrix of second partial derivatives of the cost function J(a). q is the scalar distance along the search direction specified by H and V J . The proposed method uses the Broyden-Fletcher-Goldfarb-Shanno (BFGS) algorithm [26] to build up iteratively an approximation to the inverse Hessian, and hence the vector direction. Having obtained this vector direction, the next step is to calculate 7, the distance along this direction. The technique chosen for this study is based on Brent’s method in one dimension [27].At each iteration of the nonlinear optimisation, the linear parameters are obtained as follows. Define 8, = [ U l J ,U 2 J 7 . . . , U ( T + l ) , ,bo, , blJ > . . . bs31 %+, =

-

7

$;

(4= PJ ( 4 ( W T ( i )

j = (1,2, . . . , Ad)

(11)

and

0 = [ & , 8 2 , . . . ,e,]

T

@’(i) = [ + ~ ( 2 ) 7 ~ ~ ( 2 ) 7 . . . 7 + ~ ( 2 ) 1 (12) Then one can write

The linear parameters 0 are then obtained using the Moore-Penrose pseudoinverse of Y via SVD as

o=q+.y (14) This technique is a simplification of the method due to Golub and Pereyra [28], and is therefore an approximation to the true cost function gradient. In their paper, the exact formulation for the gradient of nonlinear least squares cost functionals where the variables separate, is given in terms of the derivative of the pseudoinverse itself. The main focus of the present study, however, deals with dynamic modelling in which the parallel error cost function gradient is obtained numerically (see following Section), thus avoiding any such analytical approximations. 3.2 Optimisation of dynamic model Minimisation of the cost function given by eqn. 7 is a static optimisation of the series-parallel model (one step into the future) of the SISO plant given by eqn. 4. 507

Following usual system identification practice, the LM network is trained on one data set (training set) and tested on another data set (test set). The number of iterations taken by the optimisation method through the training data is also important when noise is present on the measurements. As training progresses, the LM network will learn the dynamic behaviour and then begin to fit the noise. This is termed overtraining or training interference. This problem can be alleviated by cross-validation using the test set [ 121. Training should be stopped at the minimum value of the test set error, since at this point the network has its maximum predictive capability. The training set error will always decrease owing to the optimisation process. Also, the predictive capability can be checked more thoroughly by forming the test set error on the parallel model output rather than the one-step-ahead error as used for training, i.e.

techniques. A schematic of the experimental pH neutralisation plant, installed at the University of California at Santa Barbara is shown in Fig. 1. This is effectively a continuous stirred tank reactor (CSTR) into which acid, base and buffer streams are added and mixed. The acid and base flowrates 41 and q3, respectively, are controlled using industry standard valves under PI control, with the buffer flowrate q2 set using a peristaltic pump. The effluent flowrate 44 depends on the fluid level in the tank and the manual valve position. The pH of the effluent is measured at a distance from the plant, introducing a measurement time delay, Td. The nominal parameters for this system are summarised in Table 1. The simulation used in this work is described in greater detail in [16]. peristaltic pump

buffer

471mixer

.........

&w

q 3 set

f x k + 1) = F ( l i ( k ) ,

4 ( k ) = [!Xk),ixk- l ) , - , i X k - d ,

4(W

c4(k) (15) This strategy, however, can be a lengthy process. The ultimate goal is, of course, to obtain a good parallel model, but the static optimisation strategy only minimises the one-step-ahead error. From a dynamic modelling point of view this can emphasise high-frequency components, resulting in overfitting and noise learning. The obvious solution therefore is to minimise the parallel model error directly. The analytic expression for the parallel model error gradient, however, is cumbersome since it involves repeated partial differentiation of the recursive expression for the network output. The complexity of the expressions obtained will increase depending on how many steps into the future the parallel prediction is performed. An alternative method is to obtain a numerical approximation to the parallel cost function gradient using forward finite-difference calculations as [26]

The finite difference in the variables should be small enough to ensure that the gradients are sufficiently smooth. Usually they are set to a small percentage of the actual parameter plus a small offset. Whereas static optimisation of the one-step-ahead error is influenced by high frequency dynamics, the reduction of the parallel error can emphasise the lowfrequency dynamics in a signal. The strategy adopted in this study proposes using the one-step ahead error to calculate the linear parameters and then forming the parallel error with the nonlinear parameters. Two different training sets can be used in this instance (one for the static model and the other for the dynamic model), further enhancing the optimisation method. Structuring the process in this way provides a compromise between learning low- and high-frequency dynamics. This should ensure that a valid model will result. odelling of p# neutralisation plant

The neutralisation of pH represents a highly nonlinear process, and hence offers a suitable case study for the demonstration and evaluation of local model network 508

valve

Fig. 1

USCB p H neutralisation plant

Table 1: Nominal operating conditions of USCB pH neutralisation plant Acid stream

0.003 M HN03

Buffer stream

0.03 M NaHC03

Base stream

0.003 M NaOH 0,00005 M NaHCO,

Acid flowrate q1

16.5ml/s

Buffer flowrate q,

0.55ml/s

Base flowrate q3

15.7 ml/s

Time delay Td

0.5 m i n

Liquid level i n tank h

12.0cm

Tank c.s.a.

207cm2

The manual effluent valve was adjusted to give the desired steady-state nominal fluid level in the reaction tank. In this particular case study the acid and buffer flowrates were kept constant at their nominal values, with the base flowrate manipulated to excite the plant dynamics. Separate training and test sets were generated using variable step changes in base flow rate. The first two sets (training set 1, test set 1) consisted of large step changes covering the entire range of operation. Although this method of nonlinear perturbation is acceptable in simulation, it is highly unlikely whether such extremes would be tolerated in practice. The third set (training / test set 2) therefore consisted of small localised perturbations to ensure linearity in different regions of the operating space. This again produces sufficient training and test data for the local model network, but the less extreme perturbations are more likely to be acceptable in practice. For this particular plant the static process gain between base flow rate and pH (i.e. the slope of the titration curve) varies considerably as the pH and base flow change (Fig. 2). It can be seen from this titration IEE PYOC-Control Theory Appl, Vol 144, No 6,November 1997

3'

10

0

20

40

30

50

60

I

70

time, min

Fig.4 p H scheduled local model network parallel model outputfor traini -p

set I

...,.,..... plant

___ Imn

2'

0

5

10

15

20

10 9-

I 30

25

a-

base flow rate, ml/s

Fig.?. p H nominal titration curve

curve that there are five regions in which the gain is fairly constant. Also, within each region, linear identification experiments suggested that a second-order dynamic model was sufficient to describe the behaviour of the process. A LM network with five local secondorder linear ARX models was therefore constructed, i.e. r = s = 1, M = 5 in eqn. 5 to give

20

0

40

60

80

100

140

120

time. min

Fig.5 p H scheduled local model network parallel model output for training/test set 2 ....

~

.....

plant Imn

5

2=1

,$( k ) = bfw)> P H ( k - 1)7 q3 - 4 q3 - 1- 41T 4 ( k ) = [ P H ( k ) Q>3 ( k- 1 - 41T (17) where pH(k) is the process output, q3(k)is the process input base flow rate at time k, respectively, and d is the system delay. In this instance, both pH and base flow rate were used to specify the operating point. A second LM network was also trained using pH alone as the operating point interpolation variable. The titration curve was then used to initialise the nonlinear activation regions. Both models were subsequently optimised using the hybrid dynamic method of Section 3. Figs. 35 and 9 show the resultant parallel model output over the three different traininghest sets and the associated interpolation functions for pH scheduling. Excellent predictions are obtained for all the test sets. Figs. 6-8 and 10-14 show the parallel model output over the same data sets and the associated interpolation functions for pH/base scheduling. In this case it is apparent that scheduling on pH and base flow rate yields improved parallel model accuracy, especially over the longer test set. 7

3,'

10

20

30

40

50

60

70

'

3' 0

IEE Proc.-Control Theory Appl., Vol. 144, No. 6, November 1997

30

40

50

70

60

I

time, min

test set 1 plant ~

Imn

20

20

30

40

50

70

60

time, min

Fig.7 pH/base scheduled local model network parallel model output for training set I

~

plant Imn

' 0

time,min plant Imn

20

Fig.6 pHbase scheduled local model network parallel model output for

20

40

60

80

100

120

140

time, min

Fig.3 p H scheduled local model network parallel model output for test set I ~

10

Fig.8 pHbase scheduled local model network parallel model output ~

.... ...... plant Imn

509

U

0

2

4

6

Fig. 13 Interpolation region for local model 4

8101214

QT!J PH d

0.8

0.4

Q

0.2

00

2

4

0.4

6PH8 1 0 1 2 1 4

0

e

Fig.9 Interpolation regions

12

a Local model 1 b Local model 2 c Local model 3 d Local model 4 e Local model 5

Fig. 14 Interpolation region for local model 5

5

Nonlinear internal model control

The basic linear internal model control (IMC) structure is given in Fig. 15. The key characteristic of this control design approach is the inclusion of a plant model within the control structure. Using this internal model an augmented feedback signal is generated which includes the effect of disturbance and, importantly, plant mismodelling. Within this general structure the various design aspects of control, such as feedback and feedforward compensation are inherently dealt with. Likewise, it provides a suitable design structure for the development of robust control techniques, owing to the presence of the mismodelling feedback signal. The filters F(z) and C(z) are designed to balance robustness to model uncertainty with disturbance response [29].

Fig. 10 Interpolation regionfor local model 1

robustness filter

inverse controller

$4 model

Fig. 11 Interpolation regionfor local model 2

robustness filter

Fig. 15 Internal model control

Initially neglecting the robustness filters, the transfer function of the plant under IMC can easily be derived as follows:

0.8 Q

0.4

ypo= d(2)

0

1 - G,(z)G,(z)

1

+ G c ( z ) ( G p ( z-) G,(z))

12

Fig. 12 Interpolation region for local model 3 510

From eqn. 18 it is apparent that perfect modelling amounts to feedforward control for setpoint tracking, IEE P r o ~ C o n t r o lTheory Appl., Vol. 144, No. 6, November 1997

with feedback control used for disturbance rejection. Likewise, it can be shown that with perfect modelling, stability of the overall structure is assured if the plant GJz), and the controller G,(z), are stable. Perfect control is obtainable from this structure if the controller G,(z) is chosen as the inverse of the internal model Gm(z). Likewise, offset-free response can be achieved if the steady-state gain of the controller is equal to the inverse of the steady-state gain of the model. In practice, perfect control cannot be achieved owing either to model zeros outside the unit circle or to the time delay present within the plant model. Naturally, this includes the delay of one sample inherent within all discrete time control systems. In such cases, the controller is designed as the inverse of the minimum phase component of the plant model. Economou, Morari and Palsson [15], have shown by the application of nonlinear operator theory that the IMC structure can be readily extended to nonlinear systems and indeed specify that the key characteristics of IMC described also apply in this nonlinear case. The approach is to include a nonlinear plant model, determined possibly from a physical understanding of the plant in place of the linear model. Invertibility conditions for this nonlinear model are discussed, with two algorithms, successive substitution and Newton’s method, suggested for the numerical inversion of this model at each sample time. A number of researchers have suggested using neural networks to provide the nonlinear plant model necessary for IMC from input/output data [16, 30, 311. In many cases the tendency is to employ a series-parallel model of the plant. However, this is not in keeping with the original linear IMC structure and it is only the parallel model of eqn. 15 that can provide offset-free control [15]. Likewise, the application of neural networks to the inverse modelling of nonlinear systems has been common in the literature, particularly in the field of robotics [32, 331. In the spirit of the IMC framework, the neural network should form a nonlinear approximation to the inverse of the previously trained neural model of eqn. 15. Assuming that the desired model output is yde,(k), then the general nonlinear inverse for the model of eqn. 15 is defined in eqn. 19. Assuming that the closed-loop system under neural IMC was stable, there would then be zero offset for asymptotically constant inputs u(k) and v(k) if the steady-state operator of the inverse model equalled the inverse of the steady-state model operator =

+ l),$(k), ..

-

A(r+I)y(k - T ) - B l ~ ( k 1) - .

-

B,u(k - s ) )

(20) This is only possible, of course, if the operating point does not depend on u(k), i.e. it must be affine in u(k), as in this case (eqn. 17). In IMC, v(k) replaces y(k + 1) (since this is not available at time k ) by the filtered setpoint response as

y(k

+ 1)

u ( k ) = F I M C ( X p l ) ( y ~ p ( k-) d ( k ) )

where d(k) = y J k ) ym(k).In this case the IMC filter FIMXz-l) was chosen to be a first-order exponential filter. Hence, only one design parameter c is needed. This is acceptable since the pH plant is of relative degree 1. For plants with a higher relative degree, a more complex filter will be needed [29]. If the controller takes the preceding form, offsets will be eliminated as follows. Consider the following steady-state values for asymptotically constant setpoints: em = lim ( r ( k )- y,(k)) ~

k+m

T,

= lim ( r ( k ) ) ktoo

If the filter gain = 1,

+

U, = r , Ymco - Ypoo therefore the steady-state error is

d4)

4 ( k ) = [$des@

5.1 Local model nonlinear IMC In the novel approach presented in this paper, the advantages of linear IMC are maintained without the problematic aspects of finding the nonlinear inverse. If the LM network has local ARX models, the exact inverse can be obtained analytically as follows. Consider the output of the LM network given by eqn. 1. If the local models A are defined as a linear ARX type, the output of the local model network can be written as an nonlinear ARX model, i.e. a NARX model as in eqn. 5. The parameters A , Biof the NARX model thus depend on the operating region. Based on this NARX model formulation, the following exact inverse control law can be postulated:

> $ ( k- r ) , U(k-

l),. . . , U ( k - s)]T

ecc = Um

- Ymcc

(23)

(24)

From the model equation

(19) To allow for a simpler computation of the inverse, less general model structures have been proposed, in particular a control-affine structure [5], or the splitting of the nonlinearity into a combination of a nonlinear function of past outputs and a nonlinear function of past inputs [ 311. Alternatively, given the general nonlinear model of eqn. 15, it is possible to solve the inverse at each successive sample time using numerical approaches such as Newton’s method [16], or the method of successive substitution [30]. IEE Proc-Control Theory A p p l , Vol. 144, No. 6, November 1997

511

Hence from eqns. 25 and 26, ymoo = U ,

-

+ em = 0

(27) Economou and Morari 1151 have also extended the stability proof for nonlinear IMC. In the usual way they proved that if the open-loop plant is stable, and if the nonlinear model is perfect and inverse stable, the closed-loop system will also be stable if the controller is the exact inverse of the model. Since the LM network consists of linear subsystems we may intuitively guess, based on the definition of the basis functions, that the entire nonlinear model is stable if the consituent linear models are stable. Around any operating point the LM network can be viewed as an equivalent linear model, but this analogy is not valid if the operating point is changing. Overall stability can be proven, however, by the use of Lyapunov's direct method. Tanaka and Sugeno [34] have used this method to prove the stability of a fuzzy model. Since the LM network is functionally equivalent to the Tagaki-Sugeno fuzzy model [33], this method is directly applicable as follows (theorem 4.2 in WI): The equilibrium of a LM network (fuzzy system) is globally asymtotically stable if there exists a common positive definite matrix P for all the subsystems such that

1 -{Dlu(k

Do

- 1)

+ Dzu(k - 2) .

*

*}

(30) Alternatively, consider a simpler method for secondorder local forward ARX models, in which the zeros can be shifted to minimum-phase positions by the simple transformation of [boilas

Since all of the [boil are known, a lower limit on R is calculated as

R > max[lboz/(lbitI - Ibotl)l (32) To maintain the forward and inverse model gains, [boi] and [bIi] are therefore replaced as

(33) This transformation has minimal effect on the forward process model, but renders the system minimum-phase and hence inverse stable.

ATPA2- P < 0 for i E { 1 , 2 , . . . , m }

(28) Here, the A, are the control canonical forms of the individual local ARX models. The local models of the pH plant are 1.0452 -0.2226 0 1.0309 -0.1952 0 1.0948 -0.2115 0 1.6683 -0.7369 0 1.1419 -0.2651 0

I

1 1 1 1 1

Q

I

6'

for which a common positive definite P is

P=

[

1 -0.82

-0.82 0.73

1

Hence, the entire LM network is stable. If, however, any of the constituent ARX models are nonminimum phase, the local model network is inverse unstable. There are a number of methods available to deal this problem [36]. One method involves factoring B(2-l) into a minimum-phase component C(z-') consisting of p zeros inside the unit circle, and a non-minimum-phase component D(z-') consisting of n - p (where n is the order of the model) zeros outside the unit circle. Only the minimum-phase component is inverted with the steady-state gain then adjusted to achieve the following IMC controller:

a(&) = c ( z - 1 ) D ( z - l )

i=O 512

13' 0

I

10

20

30

40

50

60

70

time, min

Fig. 16 Setpoint tracking, linear IMC/pH L M N IMC

_-_~

setpoint linear IMC local model IMC

5.2 Nonlinear local model network IMC control of pH plant The IMC strategy was applied to the control of the pH neutralisation plant. Fig. 16 compares the setpoint tracking behaviour of the LM network IMC (scheduled on pH, R = 0.05, c = 0.955) with a linear IMC system (the performance of the linear IMC is comparable to that obtained by a PID controller tuned to give a compromise between disturbance rejection and setpoint following for this particular plant, but results are omitted IEE Pvoc -Control Theory A p p l , Vol 144, No 6, November I997

for brevity). The results demonstrate the superior performance of the LM network controller compared to its fixed gain counterpoint. Fig. 17 shows the disturbance rejection capabilities of the LM network controller for an unmeasured disturbance of the buffer flow rate from 0.55 to Oml/s. This reduction significantly increases the process gain and causes the oscillatory behaviour of the fixed gain controller. The LM network controller, on the other hand, has a significantly better response with minimal control effort. A further comparison can be made with the nonlinear neural controller developed by Nahas et al. [16]. Despite its much simpler formulation, the LM controller performs better than the neural controller during the same disturbance and setpoint tests.

6'

18

17 -

r-

-.- -

, -

time, min

Fig. 18 Setpoint tracking, p H LMN IMCkHhase LMN IMC _ _ _ _ setpoint _ ~

_ .

LMN IMC (pH) LMN IMC (base,pH) 7.2

0

5

10

15

20

25

30

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Fig. 17 Disturbance rejection, linear IMC/PH L M N IMC _ - _ _ ~

linear IMC local model IMC

Scheduling the control on pH alone may not be the best strategy, since there may be several different pH values for each base value in the dynamic trajectory. Using both pH and base flow gives a 2-D interpolation and hence should lead to better control. Figs. 18 and 19 show that this is indeed the case, with significantly improved setpoint tracking and disturbance rejection for the LM network IMC scheduled on base flow and pH (R = 0.05, c = 0.935). The pH response during setpoint tracking is symmetrical, but the base flow-rate response is quite different for the different setpoints, indicating that the controller is reacting to the nonlinearity of the plant. 6

Conclusions

An offline training technique for a class of LM networks has been proposed which combines SVD estimation of the parameters of linear models and Hessianbased optimisation to identify the nonlinear interpolation functions. For modelling of nonlinear plant, it is IEE Proc -Control Theory A p p l , Vol 144, N o 6, November 1997

15.4

I

0

5

10

15

20

25

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Fig. 19 Disturbance rejection, p H LMN IMC/pH/buse LMN IMC _ _ _ _ LMN IMC (pH) __ LMN IMC (base, pH)

suggested that the multistep ahead or parallel prediction error be minimised directly. A finite-difference, partial-gradient approach is used for evaluating the cost function gradient, obviating the need for calculation of a highly complex, recursive dynamic gradient. 513

With ARX local models, the LM network is itself an ARX representation and is therefore readily incorporated within an internal model control framework since the the required model inverse can be derived analytically. Analysis confirms the stability of the resulting control scheme and its setpoint tracking accuracy. Simulation results for a pH neutralisation plant illustrate the potential offered by LM networks for nonlinear identification and for nonlinear internal model control. Significantly improved performance in terms of setpoint tracking and disturbance rejection compared with linear models was observed. Current research is aimed at extending the training of LM networks for online modelling and control applications. 7

Acknowledgment

Grateful acknowledgment is given for the financial support of the UK Engineering and Physical Sciences Research Council under grant GRlK69476. 8

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