Linear predictive control based on approximate input-output feedback linearisation

H.A.B.te Braake, M.Ayala Botto, H.J.L.van Can, J.Sa da Costa and HBVerbruggen

Abstract: The computational burden related to model-based predictive control (MBPC) of constrained nonlinear systems hampers its real-time application. To avoid this, input-output feedback linearisation (IOFL) techniques are used to linearise the process model over a wide operating range. The resulting linear model is then integrated in a linear MBPC scheme allowing for standard linear control techniques to be applied. However, the process input constraints become nonlinearly related with the optimisation variable due to the state-dependent nonlinear feedback law. In this paper a new method to IOFL of general multivariable discrete-time systems is proposed. By adopting an approximate IOFL based on a suitable linear model approximation, a new linear and state dependent input mapping is obtained which further enables the MBPC solution to be found through a single quadratic programming optimisation. The performance of this new technique is compared with other well-known schemes for the control of a Van der Vusse chemical reaction taking place in a CSTR.

1 Introduction

Model-based predictive control (MBPC) is a very attractive way for developing and tuning controllers and has become an important research area of automatic control with several reported successful industrial applications [ 1-31. The ability to handle process constraints in an optimal way is one of the main reasons for this success. The most common MBPC scheme is based on a linear process model, and therefore can only be used around a certain, prespecified, operating point of the nonlinear process. Linear MBPC has some important advantages. If no input, output or state constraints are present an analytical solution to the control problem exists. If constraints are present, the control problem can be rewritten as a quadratic program (QP), which can be solved in an optimal way within a finite number of steps. Unfortunately, the incor- poration of a nonlinear process model into MBPC (nonlinear MBPC) results in a nonlinear nonconvex opti- misation problem, which is only solved through time- consuming algorithms, often leading to suboptimal solu- tions. However, standard linear control techniques with reduced optimisation complexity can be applied if an equivalent linear process model is available. One of the

0 IEE, 1999 IEE Proceedings online no. 19990363 DOI: 10.1049/ip-cta:l9990363 Paper first received 4th February 1999 H.A.B. te Braake is with Heineken Technical Services, Burgemeester Smeetsweg 1, 2382 PH., Zoetenvoude, The Netherlands M. Ayala Botto and J. Sa da Costa are with the Department of Mechanical Engineering GCAWIDMEC, Instituto Superior Ttcnico, Avenida Rovisco Pais 1049-001 Lisboa, Portugal H.J.L. van Can was with Delft University of Technology, Kluyver Laboratory for Biotechnology, Julianalaan 67 2628 BC, Delft, The Netherlands, and is now with Melkunie Woerden, The Netherlands. H.B. Verbruggen is with the Delft University of Technology. Faculty of Information Technology and Systems Department of Electrical Engineer- ing, Julianalaan 67. 2628 BC, Delft., The Netherlands

IEE Proc.-Control Theory Appl., Vol. 146, No. 4, July 1999

most attractive linearisation techniques is feedback linear- isation which provides, under some mild assumptions, and by means of a nonlinear feedback, an exact linearisation of the process model in a neighbourhood of the nominal operating point [4]. The development of exact feedback linearisation techniques for nonlinear control problems has gained interest in the control engineering field. In the past decade, there have been major developments in nonlinear systems theory using techniques from differential geome- try [5-81, applied to the feedback linearisation of nonlinear systems over a wide operation range.

In this paper a new solution to the input constrained MBPC based on the input-output feedback linearisation (IOFL) of a general nonaffine multivariable discrete-time system is proposed. The main idea consists of performing a two-step IOFL procedure: first a suitable linearisation of the nonlinear prediction model over the future time steps is performed, and then an IOFL is applied to the resulting model. The main advantage OP this approximate IOFL scheme is that it leads to a linear input mapping which helps a further integration of input constraints in the MBPC framework. It is shown that the overall constrained optimisation problem can then be solved at each control sample through a single application of a QP algorithm. The comparison of the resulting trade-off between performance degradation and time consumption will be made against the former known techniques by considering a nonlinear third-order exothermic Van der Vusse reaction in a CSTR. Since neural networks are universal approximators repre- senting a continuous differentiable mapping between input and output, these input-output discrete-time models will be used to describe the dynamics of the process.

2 Input-Output feedback linearisation

Input-output feedback linearisation (IOFL) is a method to find a static state feedback control law Y, such that the resulting closed loop system has a desired linear input- output behaviour. This procedure is shown in Fig. 1 for a

295

1 X '

J x Fig. 1 Principle of input-output feedback linearisation

general nonlinear process, as the relation between y and U

is nonlinear while y is linearly related with the newly created external signal v. In general terms, the input- output feedback linearising control law Y describes a nonlinear and state-dependent relation between the process input U and the external input v, according to:

U = Y ( x , XI, v) (1) where x and XI represent the state vector information from the process and from the desired resulting linear system, respectively. Throughout this paper it is assumed that a discrete-time model of the process is available. Consider the following state-space description of a stable multi-input multi-output (MIMO) discrete-time nonlinear system:

(2) xk+l = f ( X k 3 u k ) { Y k = h(xk)

where x E X c R" is the state vector, U E U C R" is the input vector and y E Y c Rp is the output of the system, while f and h are smooth vector fields. Assume throughout that the system is square (i.e. p = m). Now, define E ( x ~ , uk) as the discrete decoupling 07 x p ) matrix given by:

= I I (3)

where h is defined according to [9]: 'i

hO[xkl = h(Xk)

h ' [Xk] = hJ-' '[ f ( x k , Uk)] v, = 1, . . . (4)

while rl , . . . , rp represent the relative degrees of each of the system outputs. If the feedback linearising control law, Y , is obtained by solving the following equation in respect to uk [4]:

E ( X k , u k ) = c A r X i + C A ' - ' B V k + C A r P 2 B V k + l

+ . . + C B v k + , - l ( 5 )

the closed-loop behaviour will be described by the input- output decoupled linear system given as:

Yk+r = c A r X L + C A r P 1 B V k + C A r - 2 B V k + l

+ . . . + CBV~+~-I (6)

Notice that eqn. 6 can represent any MIMO stable linear dynamic relation between the process output vector Y k + r and the external input vector vk = (v l ,k , . . . , vp,k)T by means of a proper choice of the linear state-space matrices A. B, and C. Typically, an analytical solution for u k in eqn. 5 cannot be easily found for general nonlinear processes. For

296

the continuous-time case, numerical methods [6] or the so- called 'extended system' method [4, 6, lo] can be helpful. Unfortunately, the extended system method cannot be applied to feedback linearise a general discrete-time system. Still the first approach is applicable to discrete- time systems; however, this involves the implementation of a time-consuming search algorithm. Therefore, two simpli- fied solutions to solve eqn. 5 will be described in the next Sections.

2. I Exact IOFL of affine discrete-time systems In one special case the input-output feedback linearisation of a discrete-time system can be calculated in a straightfor- ward way. This happens for a square discrete-time affine nonlinear input-output system, with p inputs,

with relative degree r equal to 1 for all outputs (without loss of generality), described by:

U k = ( U l . k , . . . ,up,k)? and P outputs, Y k = O i l , k , . . . ,Yp,k)',

Y k + l = f ( x k ) + G(Xk)Uk (7)

where Y k + 1 = b I , k + , . . . , yp,k+ 1): the state consists of delayed process inputs and outputs defined as x k =

with matrix G invertible for V x E X . @ l , k , . . . , Y / , k - n l , . . . ,Yp,k - n,J,U[,k - 1, . . . , u p & - m,>? and

Then it is obvious that choosing the control action as:

uk =P(xk$ 4) + Q ( X k ) v k (8)

withp(xk, x:) and Q(xk) known at time instant k. and given by:

P(x~, x',) = G-'(xk){-f (xk) + CAxkJ (9)

Q(xk) = G - ' ( x ~ ) C B (10)

results in the MIMO 0, x p ) linear system described by Y k + 1 = C A X : + C B V k .

2.2 Approximate IOFL of nonaffine discrete-time systems Consider the following nonaffine square input-output system with p inputs, uk = ( u ! , ~ , . . . , up& and p outputs, Y k = ( Y l , k , . . . ,yp,k)T, with unitary relative degree for all outputs (without loss of generality), given by:

Y k + l = f ( x k , u k ) (1 1) T with Y k + xk = b 1 . k 9 . . ' > Yl,

k - n,' ' ' . , y p , k ~ np> ,Ul,k - I, . ' . 3 U p , k - m,>T By performing a = b l , k + I 2 . . ' 1 Yp,k + 1) and

linear approximation of eqn. 11 at the operating point given by the old state trajectory, ( x k - u k - ,), using the Taylor's series expansion, one obtains:

Y k + l = f O + VFOhk + vEOAuk (12)

where f o represents the computed f at the operating point, VPo is the matrix containing the partial derivatives o f f with respect to each element of the state vector x k , VEo contains the partial derivatives off with respect to u k , both evaluated at the operating point, while f i x k = xk - xk - I and A U k = uk - uk - Then, the feedback linearking control law as given by eqn. 8 with:

p ( X k , X i ) = Uk-l + VE,'{-fo - V F o h k + CAX',] (13)

Q(xk) = VE,' CB (14)

will transform the original nonlinear nonaffine system given by eqn. 11 into a linear (p x p ) system described through the linear dynamics, Y k + M CAxk + C B v k , provided that VEo is nonsingular [ 111. This is only accu-

IEE Proc -Control Theory Appl., Vol. 146, No. 4, July 1999

rate enough if the higher order terms of the Taylor's expansion in eqn. 12 can be neglected.

3 Linear MBPC based on IOFL

The integration of the IOFL scheme in the MBPC strategy results with clear advantages to the simplification of the optimisation solution. The overall control scheme is depicted in Fig. 2. In this case, the criterion function to be minimised on every sampling instant k is a quadratic criterion on V = ( v k , . . . , v k + H p - 1): with H~ being the prediction horizon, given by:

I where j = b k + . . . , y ' k + H ) : represents the vector with the predicted linear system outputs over H,, F = (rk+ . . . , r k + H J T is the vector with the differences between the future reference signals and the filtered modelling errors, W, is a square positive definite diagonal weighting matrix of the controller outputs, and AV=(vk - v k - I , . . . , v k + H p - 1 - v k + H p - 21T is the vector which contains the changes in the future control signal. An expansion of the linear system outputs over Hp results in:

with R, and R, being expansion matrices of the linear state-space description matrices, A , B and C. The incor- poration of eqn. 16 in eqn. 15 enables the optimisation routine of the linear MBPC to find the optimal V by solving a simple analytical expression [ 121. However, if along with the criterion function, inputs, outputs or state process constraints are to be considered, an analytical solution for the optimal control vector can no longer be found. Throughout the paper, both level and rate inequality constraints acting on the process inputs are considered, and defined as:

- U 5ii cii (17)

(18) - Au 5 A i 5

with ii = ( U k , . . . , Uk+HP-1kT and Ah = ( U k - U k - 1,

. . . , u k + H p , - - u k + H p - 2 ) . Then, the minimisation of eqn. 15 subjected to eqns. 17 and 18 can only be efficiently computed if these inequality constraints are linearly expressed in terms of the optimisation variable V . In this case, fast and reliable quadratic programming (QP) opti- misation routines can be used to find the solution of the linear MBPC [13]. However, this is highly dependent on the adopted nonlinear feedback linearising control law Y. The next Sections will explore possible ways to handle process inputs constraints of the type (eqns. 17 and IS), such that the solution of the MBPC, with either the exact or the approximate IOFL schemes previously presented, can still be obtained through a QP optimisation.

1-0 feedback linearisation - _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

MBPC

model

Fig. 2

IEE Proc.-Control Theory Appl.. Vol. 146, No. 4, July 1999

Linear model-bused predictive control bused on IOFL

3.7 The introduction of an extended version of eqn. 8 over the prediction horizon Hp results in the following nonlinear and state-dependent mapping:

Constraints handling under exact IOFL

fi = G - ' ( x k , h'){-f(xk, h') +RA') + &'(Xk, h')R,;

(19)

with ii' = ( u k , . . . , u k + H p - 2>T. Since ii' contains the opti- mal process inputs over future time steps, an implicit problem arises, since the knowledge of this sequence depends on the computation of the optimal control sequence V , which, in turn, depends on the knowledge of constraints on future time steps. Therefore, QP routines cannot be applied to the minimisation of eqn. 15 under the constraints (eqns. 17 and 18), since an explicit linear relation between ii and V is not available from eqn. 19. In the sequel, three approximation alternatives to handle process input constraints are discussed in more detail, enabling QP routines to be still applied [14].

3. I. 1. Constraints on the first move only If only the first time-step-ahead process inputs are constrained, leaving unconstrained the other Hp - 1 inputs, the optimisation problem given by the minimisation of the criterion function (eqn. 15) under this new set of constraints can be solved through QP routines. Notice that eqn. 8 expresses a linear analytical relation between u k and v k , since terms p ( x k , x;) and e(&) are known at time instant k. Therefore, the following new set of linear constraints on v k are easily obtained:

with Q(xk) = Q k and P ( q , x:) = P k , for notation conveni- ence. An advantage of this approach is that no constraint violation appears at the implementation point when a receding horizon implementation is used. However, the optimal future control sequences in V do not necessarily satisfy the process input constraints over the entire horizon Hp, if H, > 1, thus the optimum found may not be the feasible optimum, which therefore may lead to unexpected behaviour.

3.1.2. Constant constraints over the whole horizon This option is described in [ 151 and simply adopts the same linear inequality constraints relations (eqns. 20-2 1) for the complete prediction horizon. Similarly to the former case, the optimal future control sequence does not have to satisfy the future process input constraints and thus the optimum found may still be a nonfeasible solution. The advantage is that the optimisation problem remains a relatively simple QP problem.

3.1.3. Linear approximation of constraints The main idea of this procedure consists of using a linear approximation of the nonlinear inequality constraints mapping that result from eqn. 19 such that QP routines can be directly applied. This technique is hl ly developed in [16], where an adaptive procedure for correcting the constraints linearisation error guarantees that a feasible control sequence V is found for the complete prediction horizon.

291

3.2 Constraints handling under approximate IOFL The expansion of the feedback linearising control law presented in Section 2.2 over Hp leads to a linear relation between ii and V, given by:

= p ( x k 3 x i ) + & x k ) i (22)

with:

p ( x k , x i ) = i k - 1 + vE;l{-& - vF() fkk + R$k} (23)

Q ( x k ) = V E i ' R , (24)

wherejO =f (& ii) is the nested output predictions of eqn. 11 over H,, computed at the old State ( x k - 1 , i ik - I), with i i k - = ( U & l , . . . , U & - 2>T taken frop the optimisa- tion at the previous _sam;ling instant, VEO contains tbe partial derivatives off&, ii) to x k at ( x k - 1, i i k - I ) , VF'O the partial derivatives off(&, ii) to ii at ( x k - 1, i i k - I) , and Axk = x k - x k - ,. Therefore, the level and rate inequality constraints acting on the process inputs (eqns. 17 and 18) are transformed into the following linear inequality constraints on the optimisation variable V :

Q i 1 [ 4 1 - p k ] 5 5 & ' [ k - p k ] (25) x

Q i ' k - p k + i ik -11 - vk-1 5 AS

- < e i l [ z - P k + i k - 1 1 - 5 k - l (26)

with Q ( x k ) = &, and p(&, x:) =p for notation conveni- ence, and ik - = ( v k - 1, . . . , v k - Hp - 2)T taken from the optimisation solution at the previous sampling instant. Eqns. 25 and 26, together with the quadratic criterion function (eqn. 15) result in a linear MBPC controller which solution can be found through a single QP optimisa- tion at each sampling instant, resulting in a very fast implementation. Moreover, by assuming a linearised feed- back law in the IOFL scheme, there is always the guarantee that constraints violation will not occur. However, this method depends on the accuracy of the mentioned linear- isation, since a rough approximation of the exact criterion function can possibly generate nonadmissible control moves. One way to prevent large linearisation errors from occurring is to choose the operating trajectory ( x k - 1,

i i k - in eqn. 22 that resulted from the optimisation applied at the pre_vious sampling instant [ 171.

4 Control of a nonlinear chemical reactor

In this Section the input constrained MBPC, based on feedback linearisation, is applied to the control of a third- order exothermic Van der Vusse reaction taken place in a cooled CSTR. The reaction is characterised by:

2A 2 D

A model of the system is given in [ 181 and consists of three states (concentrations), one output (concentration of B) and two inputs: u1 the dilution rate of the reactor and u2 the heat exchanged between the CSTR and the environment which will be kept constant during the simulations. The control signal u1 is bounded by the following level and rate constraints (with T= 30 seconds, being the sampling time):

298

4.1 Modelling phase Two types of feedforward neural networks will be used for modelling the dynamics of the Van der Vusse reaction: a (control) affine neural network and a nonaffine neural network. The resulting discrete-time input-output models will be feedback linearised by means of an exact IOFL or an approximate IOFL, respectively (see Sections 2.1 and 2.2 for details). Both networks are trained based on measured input-output data taken from the process simu- lation, the identification data set, and validated on a 'fresh' validation data set. Both data sets are noise free, generated by superimposed sinusoidal signals for u1 covering approximately the same range in the process output values. After completing the learning phase, the following affine neural network model resulted:

where the state (regressor) vector is given by x k ' = I y k ,

Y k - 1, Y k - 2, u k - 11. The best structure found for each f, and g, in eqn. 28 consists of one hidden layer feedforward neural network having three neurons with tangent hyper- bolic activation function. Meanwhile, the best nonaffine neural network model is found to have the following structure:

with the regression vector. Xkna = b k , y k - Y k - 2, uk, u k - and having five neurons with tangent hyperbolic activation function in the single hidden layer. Both neural network configurations showed comparable prediction performance.

4.2 Control results The simulation results with the linear MBPC use the linearisation schemes presented in the paper: the exact IOFL applied to the affine neural network discrete-time model (eqn. 28), and the approximate IOFL applied to the nonaffine neural network discrete-time model (eqn. 29). The three constraints handling procedures described in Section 3.1 for the exact IOFL, as well as the procedure presented in Section 3.2 for the approximate IOFL will cope with the level and rate constraints defined in eqn. 27. In both cases, the parameters of the resulting IOFL linear model are obtained through a Jacobian linearisation of the nonaffine neural network model around a certain stable equilibrium point, and given by:

r o 0 0 0 1

I I , .Or7 1.4891 -0.6462 0.0746 1 0 0 xk+l =

L o 0 1

1 1 -0.005 1 1 + L : 1"

For the sake of comparison, the closed-loop performance of each controller is analysed with respect to two variables: the relative sum-squared output tracking error, and the relative optimisation time needed for completing the simu- lation. These measurements are taken by relative compar- ison with the results taken from the unconstrained case with the same simulation conditions. A square reference

IEE Proc.-Conirol Theory Appl., Vol. 146, No. 4, July 1999

Table 1: Controller results relative to the unconstrained case (first line: sum-squared error relative to 100, second line (italic numbers): optimisation time relative to 1)

.- 5 10 4-4 - ._ : O!

-1 0

Control Constraints Prediction Horizon(H,) strategy handling 2 3 4

... ... AU .. .......... ... .. . .. . ..,.,... ... .. ... . , ........ .,... .. -

. . . . . . . . . 1 ............. u..? ........ If . . . . . . . . . ................

5 6 7 8 9 10 20

linear MBPC based on exact IOFL

(using affine

neural network)

linear MBPC based on approx. IOFL

(using nonaffine

neural network)

nonlinear MBPC (using nonaffine

neural network)

constraint the first move only constant over whole horizon linear approx. whole horizon

linear over whole horizon

linear over whole horizon

947.3

1.8

947.4

1.7

946.8

4.7

946.2

3.5

1073.3

6.6

748.8

1.8

748.8

1.8

748.0

7.1

747.8

4.8

764.8

10.8

777.2

1.9

594.6

1.9

593.9

9.7

593.6

6.2

661.4

18.3

783.8

2.0

51 1.9

2.1

509.1

12.3

51 1.4

7.6

608.0

29.7

782.7 782.4

2.1 2.1

434.7 435.9

2.3 2.5

433.6 418.8

15.2 18.2

433.7 418.3

9.3 11.1

445.8 485.5

35.8 54.4

782.4 782.3

2.2 2.3

404.1 392.5

2.7 3.1

396.9 383.4

21.4 25.0

397.2 383.9

12.9 15.0

533.7 408.2

79.7 110.5

782.2 782.0

2.4 3.7

399.7 442.7

3.4 8.9

378.9 378.8

28.5 72.9

378.7 378.3

16.9 44.6

506.6 413.5

127.3 1200

tracking problem for over three hours is considered, where a 25 J/s disturbance is added to the heat exchange u2 during a period of 25 minutes (from 1 h 15 min till 1 h 40 min). No noise is added to the Van de Vusse simulation model, a simple unitary static filter is used to feedback the

30r

01 I

300 400 500 600 700 800 900 1000 relative sum squared error

Fig. 3 solid line: approx. IOFL with linear constraints, dotted line: exact IOFL with linear approx. of constraints

Comparison of linear MBPC performance for Hp = 2 , . . . , 10

1 .I 1 n

0.7

0.6

0 5

c

0 20 40 60 80 100 120 140 160 180 time, min

Fig. 4 Hp= 10 solid line: process output, dotted line: reference signal

IEE Proc.-Control Theory Appl., Vol. 146, No. 4, July 1999

Simulation resultsfrom linear MBPC based on approx. IOFL with

modelling errors (see Fig. 2) , while a weighting factor of 2.5 x 10 - is used for the diagonal matrix W, in eqn. 15 and kept constant for all controller configurations. The simulation results presented in Table 1 show the perfor- mance of each controller configuration for various predic- tion horizons. The simulation results concerning the application of the nonlinear constrained MBPC using the nonaffine network model are also presented in last line of Table 1. Using this controller structure, a nonconvex, nonlinear optimisation problem has to be solved by numer- ical search routines, despite a straightforward handling of the inequality constraints. The inevitable local solutions and the total computational time requirements are major problems which are emphasised as the dimensionality of the optimisation problem increases. The comparison between the two configurations which guarantee a feasible control solution over the complete prediction horizon, namely the exact IOFL with linear approximation of constraints and the approximate IOFL, show a similar performance in terms of the closed-loop tracking error, although the approximate IOFL is less computational time demanding, as shown in Fig. 3. The simulation results in Figs. 4 and 5 indicate a good and stable tracking perfor-

H -10 s& line: control action dotted line: control bounds

299

mance, proving as well that both level and rate input process constraints are obeyed for the whole simulation.

5 Conclusions

This paper formulates the combination of two methods to input-output feedback linearise multivariable discrete-time systems with constrained MBPC, such that a feasible solution is always found through a QP optimisation. The exact IOFL uses an affine model of the process and requires a constraints handling approximation procedure to guarantee the feasibility of the iterative QP optimisation. In contrast, the approximate IOFL is based on a more general nonaffine model which, due to its approximate nature, leads to a feasible control solution through a single QP optimisation. Despite the approximation errors here involved, results for the Van der Vusse reactor show similar closed-loop performance for both methods with a smaller computational requirement when using the approximate IOFL.

Further research is actually being directed towards the study of the stability and robustness properties of the overall closed-loop system. One of the most interesting features of the approximate IOFL scheme, is the possibility to extend the stability robustness and performance-robust- ness properties of the well-known linear pole-placement control to the general nonlinear case [19, 201. In the author’s opinion, the resulting framework will provide a step forward on the development of robust nonlinear controllers.

6

1

2

References

BRISK, M.L.: ‘Process control: theories and profits’, Proceedings IFAC world congress, Sydney, Australia, 1993, 7, pp. 941-950 EDER, H.H.: ‘Predictive control in practice: benefits and barriers’, 1 A , 1994,35 (3), pp. 16-17

3

4

5

6

I

8

9

10

11

12

13

14

15

16

17

18

19

FUCHALET, J.: ‘Industrial applications of model based predictive control’, Automafica, 1993, 29, pp. 1251-1274 NIJMEIJER, H., and VAN DER SCHAFT, A.J.: ‘Nonlinear dynamical control systems’ (Springer-Verlag, New York, 1990) KRAVARIS, C., and KANTOR, J.C.: ‘Geometric methods for nonlinear process control, Part 1, background. Part 2, controller synthesis’, Ind. Eng. Chem. Res., 1990,29, pp. 2295-2323 HENSON, M.A., and SEBORG, D.E.: ‘Input-output linearization of general nonlinear processes’, AIChE, 1990,36, vol. 11, pp. 1753-1757 HENSON, M.A., and SEBORG, D.E.: ‘Critique of exact linearization strategies for process control’, 1 Proc. Control, 1991, 1, pp. 122-139 EL MOUBARAKI. J.. BASTIN. G.. and LEVINE. J.: ‘Nonlinear control of biotechnological processes’ with growth-production decoupling’, Math. Biosci., 1993, 116, pp. 2 1 4 1 MONACO, S., and NORMAND-CYROT, D.: ‘Minimum-phase nonlinear discrete-time systems and feedback stabilization’, Proceedings of the 28th IEEE conference on Decision and Control, Los Angeles, USA, 1987, pp. 979-986 VAN DER SCHAFT, A.J.: ‘Linearization and input-output decoupling for general nonlinear systems’, Syst. Cont. Lett., 1984, 5, pp. 27-33 TE BRAAKE, H.A.B., VAN CAN, H.J.L., SCHERPEN, J.M.A., and VERBRUGGEN, H.B.: ‘Control of nonlinear chemical processes using dynamic neural models and feedback linearization’, Comp. Chem. Eng., 1998, 22, pp. 11 13-1 127 CAMACHO, E.F., and BORDONS, C.: ‘Model predictive control in the process industry’ (Springer-Verlag, London, 1995) GILL, P.E., and MURRAY, W.: ‘Numerical methods for constrained optimization’ (Academic Press Inc., London, 1974) NEVISTIC, V, and MORARI, M.: ‘Constrained control of feedback- linearizable systems’, Proceedings of 3rd European control conference, Rome, Italy, 1995, pp. 1726-1731 HENSON, M.A., and KURTZ, M.J.: ‘Input-output linearisation of constrained nonlinear processes’, Session on Nonlinear Control, AIChE annual meeting, San Francisco, CA, 1994, pp. 1-20 AYALA BOTTO, M., VAN DEN BOOM, T.J.J., KRIJGSMAN, A., and SA DA COSTA, J.: ‘Predictive control based on neural network models with VO feedback linearization:, Int. 1 Control, 1999 AYALA BOTTO, M., and SA DA COSTA, J.: ‘A comparison of nonlinear predictive control techniques using neural network models’, 1998,J Syst. Architecture, 44, pp. 597-616 DE OLIVEIRA, S., NEVISTIC, V, and MORARI, M.: ‘Control of nonlinear systems subject to input constraints’, Preprints of NOLCOS’95. Tahoe Citv. California. 1995. 1. DD. 15-20

’ S0RENSEN,’ 0.: ‘No&near pole-placemeit’ control with neural networks’, Proceedings of third European control conference, Rome, Italy, 1995, pp. 2742-2147

20 WAMS, B., AYALA BOTTO, M., VAN DEN BOOM, T.J.J., and SA DA COSTA, 1.: ‘Training neural networks for robust control of nonlinear MIMO systems’, Proceedings of the international conference on Control’98, Swansea, UK, 1998, pp. 241-246

300 IEE Proc.-Control Theory Appl., Vol. 146, No. 4, July 1999