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Article

Transactions of the Institute of

Measurement and Control

35(2) 166180

The Author(s) 2012

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DOI: 10.1177/0142331212438910

tim.sagepub.com

RungeKutta model-based adaptivepredictive control mechanism for

non-linear processes

Serdar Iplikci

Abstract

This paper proposes a novel non-linear model predictive control mechanism for non-linear systems. The idea behind the mechanism is that the so-

called RungeKutta model of a continuous-time non-linear system can be regarded as an approximate discrete model and employed in a generalized

predictive control loop for prediction and derivative calculation purposes. Additionally, the RungeKutta model of the system is used for state estima-

tion in the extended Kalman filter framework and online parameter adaptation. The proposed method has been tested on two different non-linear sys-

tems. Simulation results have revealed the effectiveness of the proposed method for different cases.

Keywords

Extended Kalman filter, non-linear model predictive control, optimal control, parameter estimation

Introduction

Since the appearance of the first model predictive control

(MPC) technique (Richalet et al., 1978), many MPC methods

have been proposed (Cutler and Ramaker, 1980; Keyser and

Cauwenberghe, 1985; Soeterboek, 1992) in the literature and

they have been proven to be effective and robust tools forcontrolling industrial processes especially for non-minimum

phase plants, open-loop unstable plants and plants with vari-

able dead-time and/or parameters (Camacho, 1993; Clarke

and Mohtadi, 1989). They have been applied not only to

industrial processes (Clarke, 1988; Richalet, 1993), but also to

a wide spectrum of areas ranging from chemistry to aerospace

(Qin and Badgwell, 2003) mainly because of their superior

features, namely the time-domain formulation, receding hori-

zon scheme and constraint-handling capability. Among oth-

ers, the most special and probably the most common one is

the so-called generalized predictive control (GPC) technique

(Clarke et al., 1987a,b; Clarke, 1994). Still, all MPC tech-

niques share the same idea: the model of the plant is used to

predict future behaviour of the plant in response to a candi-

date control vector, which is to be optimized by solving a

finite-horizon open-loop optimal problem during each sam-

pling period, and then the first element of the optimized con-

trol vector is applied to the plant. In this respect, the model of

the plant plays a crucial role in the MPC methods. The first

MPC technique (Richalet et al., 1978) is based on an impulse-

response model, whereas another early method (Cutler and

Ramaker, 1980) is based on the step-response model of the

plant. Although the MPC techniques based on the impulse-

and step-response models exhibit adequate performance for

stable systems, they are inapplicable to unstable ones. In

order to overcome this problem, techniques based on the

CARIMA (Clarke et al., 1987a,b) and CARMA (Keyser and

Cauwenberghe, 1985) models have been proposed. Even if

these MPC techniques have been suggested for discrete-time

systems at the beginning, their continuous-time versions can

be found in the literature (Demircioglu and Gawthrop, 1991;

Gawthrop and Demircioglu, 1989). These GPC methods have

originally been developed for linear plants, which leads to anoptimization problem that can be solved analytically. This

study, on the other hand, concentrates on controlling non-

linear continuous-time processes. In this case, one solution is

to linearize the non-linear system dynamics around the oper-

ating points and then to apply a linear GPC technique based

on the linearized model. However, the linearization approach

may lose its efficiency in the following cases: (a) highly non-

linear processes operating in the vicinity of a fixed point, (b)

moderately non-linear processes operating in the whole oper-

ating region, (c) reference signals varying frequently in the

whole operating region. Another remedy is the so-called non-

linear model predictive control (NMPC) technique (Camacho

and Bordons, 2003, 2007; Henson, 1998), which employs anon-linear model of the plant. The NMPC technique entails

solving a constrained non-linear optimization problem during

every sampling period. It is usually a very difficult and time

consuming task to find the global solution of such a problem.

Fortunately, it is demonstrated in Scokaert et al. (1999) that

Department of Electrical and Electronics Engineering, Pamukkale

University, Turkey

Corresponding author:

Serdar Iplikci, Department of Electrical and Electronics Engineering,

Pamukkale University, Denizli, Turkey.

Email: [email protected]

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instead of finding the global solution, feasible, sub-optimal

solutions to this problem can yield a stabilizing controller.

Moreover, as stressed in Maciejowski (2002), it is even not

necessary to find a local minimum and the only task during

each sampling period is to find a solution that provides a suf-

ficient decrement in the cost function.

NMPC relies on the model of the non-linear plant and, as

discussed in Henson (1998) and Camacho and Bordons (2003),there are three types of non-linear plant models that are

employed in the NMPC loop. The first one is referred to as the

fundamental models, which are obtained by applying first prin-

ciples (e.g. mass, energy and momentum balances), resulting in

ordinary differential equations (ODEs) and in some cases addi-

tive algebraic equations. The second type of non-linear plant

models are obtained by inputoutput measurements and they

are therefore referred to as the empirical models. In the litera-

ture, there exist many types of empirical models employed in

the NMPC loop, such as polynomial ARMA models

(Hernandez and Arkun, 1993), Hammerstein models (Fruzzetti

et al., 1997), Volterra models (Doyle et al., 1995; Genceli and

Nikolaou, 1995; Gruber et al., 2010; Maner et al., 1996),Wiener models (Cervantes et al., 2003; Norquay et al., 1998),

and also models using soft computing tools, namely artificial

neural networks (Lawrynczuk, 2007; Piche et al., 2000), fuzzy

logic (Cho et al., 1999; Roubos et al., 1999), and machine

learning tools like support vector machines (Iplikci, 2006,

2010; Xi et al., 2007). Finally, the hybrid non-linear models

combine the fundamental and empirical approaches.

This work has been concentrated on MPC of non-linear

systems based on the fundamental models. More specifically,

the focus is on the continuous-time systems because a great

deal of the natural processes and industrial plants are in the

form of continuous-time systems. However, most of the

NMPC techniques are most naturally developed for discrete-

time systems, and therefore, it is usually necessary to convert

the non-linear optimal control problem (NOCP) into a non-

linear programming (NLP) problem by either discretization

or approximation methods, and then solve it using an NLP

solver. For this purpose, several methods, which are referred

to as the direct methods, have been proposed in the literature,

for instance collocation on finite elements (Kawathekar and

Riggs, 2007), multiple shooting (Schafer et al., 2007) and their

combinations (Tamimi and Li, 2010). Alternatively, another

approach is to employ the discretized models of the

continuous-time systems by discretizing their ODEs in the

fundamental models obtained by first principles. In Sistu and

Bequette (1996), for example, explicit and implicit Euler

methods have been considered.

This paper, on the other hand, proposes a novel NMPC

method for non-linear continuous-time systems, which is

based on the classical fourth-order RungeKutta algorithm.

In the method, a discretized model of the non-linear continu-

ous-time system, which is referred to as the RungeKutta

model of the system, is obtained by the RungeKutta algo-

rithm and then employed in the GPC loop. As will be demon-

strated in the next section, the RungeKutta model of thesystem is used not only for prediction and derivative calcula-

tion purposes in the GPC framework but also for state and

parameter estimation purposes. In the literature, there exist

many numerical integration methods for solving ODEs (Press

et al., 2007). The motivation behind the use of the fourth-

order RungeKutta algorithm among others is the fact that it

has been proved to be very accurate and stable and also works

well in many applications. The novelty in the paper is the use

of the classical RungeKutta numerical integration algorithm

to discretize the non-linear continuous-time system to obtain

its so-called RungeKutta model, whereby it is possible to

implement MPC, state estimation and online parameter esti-

mation, which results in an adaptive non-linear predictivecontroller.

This paper is organized as follows: divided into five sub-

sections, the next section introduces the proposed Runge

Kutta-based control scheme. The first subsection reviews the

classical fourth-order RungeKutta algorithm. The proposed

structure is given in the following subsection. In the third

subsection, it is demonstrated how to obtain future predic-

tions and Jacobian calculations using the RungeKutta

models. The RungeKutta model-based state and online

parameter estimation blocks, which are used in the proposed

structure, are discussed in the last two subsections, respec-

tively. In the section after that, the effectiveness of the pro-

posed structure is demonstrated on two different non-linear

multiple-input multiple-output (MIMO) systems. At the endof the section, total computation times for each system

have been given in order to have a basis for real-time applic-

ability of the proposed method. The paper ends with the

conclusions.

The proposed RungeKutta-based

control structure

MIMO systems and the RungeKutta method

Consider an N-dimensional continuous-time MIMO system

as illustrated in Figure 1(a). The state equations of the system

are

Figure 1 (a) A continuous-time multiple-input multiple-output (MIMO) system and (b) its RungeKutta model.

Iplikci 167

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_x1 =f1(x1(t), . . . ,xN(t), u1(t), . . . , uR(t), u)

.

.

.

_xN =fN(x1(t), . . . ,xN(t), u1(t), . . . , uR(t), u)

1

subject to state and input constraints of the form

x1(t) 2 X1, . . . ,xN(t) 2 XN, 8t 0u1(t) 2 U1, . . . , uR(t) 2 UR, 8t 0 2

Where Xis and Uis are box constraints for the states and

inputs as given below, respectively

Xi = fxi 2 < j ximin

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state of the plant and the accurate value of the system para-

meters, which are provided by the RungeKutta model-based

EKF and parameter estimation blocks, respectively.

Based on these predictions, the candidate control vector

u(n) is updated in a manner such that the sum of the squared

K-step ahead prediction errors are minimized with minimumdeviations in the control actions, thereby resulting in an opti-

mal control action u*(n)= u(n) +du(n) to be applied to the

plant. In other words, it is attempted to minimize the objec-

tive function Fgiven by (11)

F(u(n)) =XQq= 1

XKk= 1

(yq(n+ k) ^yq(n+ k))2

+XRr= 1

lr(ur(n) ur(n 1))2 11

where K is the prediction horizon and lr is the penalty term

associated with the rth input. For an optimal control actionu*(n), an additive correction term du(n) has to be found such

that the objective function F(u(n) + du(n)) is minimized,

while satisfying the condition for descent direction: F(u(n)

+ du(n))\ F(u(n)). The CFM block tries to minimize this

objective function with respect to du(n) based on the second-

order Taylor approximation as follows:

F(u(n) + du(n)) ffi F(u(n)) + F(u(n))u(n)

Tdu(n)

+1

2du(n)T

2F(u(n))

u2(n)

du(n) 12

whereF(u(n))

u(n)is the gradient vector and

2F(u(n))

u2(n)is the

Hessian matrix. Since it is sought for a du(n) that minimizes

the objective function, if the derivative of the approximation

ofFwith respect to du(n) is taken and then equated to zero as

F(u(n) + du(n))

du(n)ffi F(u(n))

u(n)+

2F(u(n))

u2(n)du(n) = 0 13

then du(n) is obtained as

du(n) = 2F(u(n))

u2(n)

1F(u(n))

u(n)14

which corresponds to the Newton direction that provides a

quadratic convergence to the local minimum if the Hessian

matrix is positive definite (for descent direction) and the higher-

order terms are negligibly small (Nocedal and Wright, 1999;

Venkataraman, 2002). If the Hessian matrix is not positive defi-nite, a judiciously chosen additive term gI can be added to the

Hessian to make this extended Hessian matrix positive definite.

By judiciously, we mean g should be chosen slightly larger

than the most negative eigenvalue of the Hessian matrix.

At this point, it is apparent that the calculation of the

gradient and Hessian terms, i.e. the first- and second-order

derivatives of the objective function with respect to u(n),

is needed. However, in order to avoid calculating the

time-consuming second-order derivatives, the well-known

Jacobian approximation can be employed (Nocedal and

Wright, 1999), which suggests that the (KQ + R)3R Jacobian

matrix J

)1(1 ny

)1(nyQ

1z

1y

Qy

])()1([ 11 Knyny

])()1([ Knyny QQ

)(*1 nu

)(* nuR

1z

1z

)(~1 nx )(~ nxN

1z

1z

1z

)]([ 1 nu

)]([ nuR

Cost Function

Minimization

Runge-Kutta

Model

Runge-KuttaModel Based

Parameter

Estimation

Runge-Kutta

Model Based

EKF

MIMO

System

Figure 2 Proposed RungeKutta-based control structure.

Iplikci 169

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J =

e(n+ 1)u1(n)

e(n+ 1)u2(n)

. . .e(n+ 1)uR(n)

.

.

....

..

....

e(n+K)u1(n)

e(n+K)u2(n)

. . .e(n+K)uR(n)

.

.

....

..

....

e(n+KQ)u1(n)

e(n+KQ)u2(n)

. . .e(n+KQ)

uR(n)ffiffiffiffil1p (u1(n)u1(n1))u1(n)

ffiffiffiffil1p (u1(n)u1(n1))u2(n)

. . .ffiffiffiffil1p (u1(n)u1(n1))

uR(n)

.

.

....

..

....ffiffiffiffi

lRp

(uR(n)uR(n1))u1(n)

ffiffiffiffilR

p(uR(n)uR(n1))

u2(n). . .

ffiffiffiffilR

p(uR(n)uR(n1))

uR(n)

266666666666666666664

377777777777777777775

=

^y1(n+ 1)u1(n)

^y1(n+ 1)u2(n)

. . .^y1(n+ 1)uR(n)

.

.

....

..

....

^y1(n+K)u1(n)

^y1(n+K)u2(n)

. . .^y1(n+K)

uR(n)

.

.

....

..

....

^yQ(n+K)

u1(n)

^yQ(n+K)

u2(n). . .

yQ(n+K)

uR(n)

ffiffiffiffiffil1

p

ffiffiffiffiffil1

p. . .

ffiffiffiffiffil1

p

..

....

. ..

..

.ffiffiffiffiffilR

p ffiffiffiffiffilR

p. . .

ffiffiffiffiffilR

p

26666666666666666664

37777777777777777775

15

can represent the gradient vector exactly and the Hessian

matrix approximately as

F(u(n))

u(n)=2JTe and

2F(u(n))

u2(n)ffi 2JTJ 16

respectively, where e is the vector of prediction errors and

input slews given by

e =

e(n+1)

.

.

.

e(n+K)

.

.

.

e(n+KQ)ffiffiffiffiffil1

pDu1(n)

.

.

.ffiffiffiffiffilR

pDuR(n)

266666666666664

377777777777775=

y1(n+1) ^y1(n+1)...

y1(n+K) ^y1(n+K)...

yQ(n+K) ^yQ(n+K)ffiffiffiffiffil1

p(u1(n) u1(n 1))

.

.

.ffiffiffiffiffilR

p(uR(n) uR(n 1))

266666666666664

37777777777777517

Thus, the correction term is computed as

du(n) = (JT

J +mI)1

JT

e 18

which implies that it is needed only the first-order derivatives

byq(n+ k)ur(n)

for q=1,.,Q and r=1,.,R.

By using the RungeKutta model of the plant given in

Equation (9), it is explained in the next subsection how to

obtain the K-step ahead future predictions of the plant outputs

and also the necessary derivatives for the Jacobian calculations.

Future predictions and Jacobian calculations

The K-step ahead future predictions of the plant outputs

can be calculated by using the RungeKutta model of the

plant if Equations (6) and (7) are employed in an iterative

manner as

x(n+ k) = f(x(n+ k 1), u(n), u)y(n+ k) = g(x(n+ k 1), u(n)) for k= 1, . . . ,K 19

It should be noted that the candidate control vector u(n) is

assumed to remain unchanged during the prediction interval

([t + Ts t + KTs]). Hence, a series of future predictions is

obtained for each output as

^yq(n+ 1), . . . , ^yq(n+K), for q=1, . . . ,Q 20

In order to obtain the necessary derivatives for the Jacobian

calculation, first, Equations (6)(8) are rewritten as

^xi(n+ k) = ^xi(n+ k 1) + 16K1Xi(n+ k 1)

+1

3K2Xi(n+ k 1) + 1

3K3Xi(n+ k 1)

+1

6K4Xi(n+ k 1) 21

for i=1,.,Nand

^yq(n+ k) =gq(^x1(n+ k), . . . , ^xN(n+ k), u1(n), . . . , uR(n))

22

for q=1,.,Q, where

K1Xi(n+ k 1) =Tsfi(^x1(n+ k 1), . . . , ^xN(n+ k 1),u1(n), . . . , uR(n), u) 23

K2Xi(n+ k 1) =Tsfi(^x1(n+ k 1) + 0:5K1X1(n+ k 1), . . . ,xN(n+ k 1) +0:5K1XN(n+ k 1), u1(n), . . . , uR(n), u)

24

K3Xi(n+k 1) =Tsfi(^x1(n+k 1) +0:5K2X1(n+k 1), . . .,^xN(n+k 1) +0:5K2XN(n+k 1),u1(n), . . .,uR(n),u)

25

K4Xi(n+ k 1) =Tsfi(^x1(n+ k 1) +K3X1(n+ k 1), . . . ,^xN(n+ k 1) +K3XN(n+ k 1), u1(n), . . . , uR(n), u)

26

Now, the problem of calculation of the derivatives is turned

out to that of the termsbyq(n+ k)

ur(n), which can be handled as

follows:

170 Transactions of the Institute of Measurement and Control 35(2)

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yq(n+ k)

ur(n)=

gq(^x1(n+ k), . . . ,xN(n+ k), u1(n), . . . , uR(n))

ur(n)

=gq

ur+

gq

^x1(n+ k)

^x1(n+ k)

ur(n)+ . . . +

gq

xN(n+ k)

^xN(n+ k)

ur(n)

x1 = ^x1(n+ k)

.

.

.

xN = ^xN(n+ k)

=gq

ur+

gq

x1

^x1(n+ k)

ur(n)+ . . . +

gq

xN

^xN(n+ k)

ur(n)

x1 = ^x1(n+ k)

.

.

.

xN = ^xN(n+ k)

=gq

ur+XNi= 1

gq

xi

^xi(n+ k)

ur(n)

" #x1 = ^x1(n+ k)

.

.

.

xN = ^xN(n+ k)

27

Here

^xi(n+ k)

ur(n)=

^xi(n+ k 1)ur(n)

+1

6

K1Xi(n+ k 1)ur(n)

+1

3

K2Xi(n+ k 1)ur(n)

+1

3

K3Xi(n+ k 1)ur(n)

+1

6

K4Xi(n+ k 1)ur(n)

28where

K1Xi(n+ k 1)ur(n)

=Tsfi(^x1(n+ k 1), . . . , ^xN(n+ k 1), u1(n), . . . , uR(n), u)

ur(n)

=Tsfi

ur+XNj= 1

fi

xj

^xj(n+ k 1)ur(n)

" #x1 = ^x1(n+ k1)

.

.

.

xN = ^xN(n+ k1)29

and

K2Xi(n+ k 1)ur(n)

=Tsfi

ur

x1 = ^x1(n+ k1) +K1X1(n+ k1)=2

.

.

.

xN =xN(n+ k1) +K1XN(n+ k1)=2

+TsXNj=1

fi

xj

^xj(n+ k 1)ur(n)

+1

2

K1Xj(n+ k 1)ur(n)

" #x1 = ^x1(n+ k1) +K1X1(n+ k1)=2

.

.

.

xN = ^xN(n+ k1) +K1XN(n+ k1)=2

30and

K3Xi(n+ k 1)ur(n)

=Tsfi

ur

x1 = ^x1(n+ k1) +K2X1(n+ k1)=2

.

.

.

xN =xN(n+ k1) +K2XN(n+ k1)=2

+Ts XN

j=1

fi

xj

^xj(n+ k 1)ur(n)

+1

2

K2Xj(n+ k 1)ur(n) " # x1 = ^x1(n+ k1) +K2X1(n+ k1)=2.

.

.

xN = ^xN(n+ k1) +K2XN(n+ k1)=2

31

and

K4Xi(n+ k 1)ur(n)

=Tsfi

ur

x1 = ^x1(n+ k1)+ K3X1(n+ k1)

.

.

.

xN = ^xN(n+k1) +K3XN(n+ k1)

+TsXNj=1

fi

xj

^xj(n+ k 1)ur(n)

+1

2

K3Xj(n+ k 1)ur(n)

" #x1 = ^x1(n+ k1) +K3X1(n+k1)

.

.

.

xN = ^xN(n+k1) +K3XN(n+ k1)

32

Consequently, the necessary derivatives can be obtained by

using the RungeKutta model of the plant.

The RungeKutta model-based EKF

This subsection describes the estimation of the current

state of the plant based on its RungeKutta model incombination with the well-known EKF approach (Grewal

and Andrews, 2008; Welch and Bishop, 2006). Before dis-

cussing the RungeKutta model-based EKF, it is helpful

to review EKF. Consider a non-linear discrete-time system

given by

x(n+ 1) = h(x(n), u(n)) + w(n)

y(n+ 1) = g(x(n), u(n)) + v(n)33

where x is an N-dimensional state vector to be estimated,

u 2

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~xi(n)

~xj(n 1) =di,j+1

6

K1Xi(n 1)~xj(n 1) +

1

3

K2Xi(n 1)~xj(n 1)

+1

3

K3Xi(n 1)~xj(n 1) +

1

6

K4Xi(n 1)~xj(n 1) 39

where

K1Xi(n 1)~xj(n 1) =Ts

fi(~x1(n 1), . . . , ~xN(n 1), u1(n 1), . . . , uR(n 1))~xj(n 1)

=Tsfi

xj

x = ~x(n1)u = u(n1)

40

and

K2Xi(n 1)~xj(n 1)

= TsXNp=1

fi

xp

1

2

K1Xp(n 1)~xj(n 1) + dp,j

" #x1 = ~x1(n 1) +K1X1(n 1)=2

.

.

.

xN= ~xN(n 1) +K1XN(n 1)=2u = u(n 1)

41

and

K3Xi(n 1)~xj(n 1)

= TsXNp=1

fi

xp

1

2


" #x1 = ~x1(n 1) +K2X1(n 1)=2

.

.

.

xN= ~xN(n 1) +K2XN(n 1)=2u = u(n

1)

42

and

K4Xi(n 1)~xj(n 1)

= TsXNp= 1

fi

xp

1

2


" #x1 = ~x1(n 1) +K3X1(n 1)

.

.

.

xN= ~xN(n 1) +K3XN(n 1)u = u(n 1)

43

Thus, the current state of the plant can be estimated by itsRungeKutta model employed in the EKF algorithm.

The RungeKutta model-based online parameter

estimation

Assume that previous state x(n) and current state x(n +1) of

a non-linear system (1) are given directly (or estimated by

EKF) at time (n + 1)Ts and that the previous control input

u(n) is known. If the RungeKutta model of the system is

employed, it is definitely possible to relate the current state of

the system to its previous state, inputs and the parameter (u)

by Equation (6). Thus, the parameter (u) of the system can be

estimated as follows: first, x(n + 1) is predicted by the Runge

Kutta model of the system and then the vector of prediction

errors related to the parameter is formed as

e =

e1e2

...

eN

26664 37775=x1(n+1) ^x1(n+ 1)x2(n+1)

^x2(n+ 1)

...

xN(n+1) ^xQ(n+ 1)26664 37775 44

Next, the Jacobian matrix consisting of the partial derivatives

of the errors with respect to the parameter is constructed as

Ju =e1

u

e2

u. . .

eN

u

T45

Finally, the parameter of the system is updated by

un+ 1 = un JTu e

JTu Ju46

In this respect, the derivatives bx i(n+ 1)u for i= 1,.

,N areneeded for calculation of the Jacobian, which can be obtained

by using Equation (6) of the RungeKutta model of the sys-

tem as described below.

^xi(n+ 1)

u=

^xi(n)

u+

1

6

K1Xi(n)

u+

1

3

K2Xi(n)

u

+1

3

K3Xi(n)

u+

1

6

K4Xi(n)

u47

where

K1Xi(n)

u=Ts

fi(^x1(n), . . . , ^xN(n), u1(n), . . . , uR(n))

u

=Tsfi

u

x = x(n)u = u(n)

48

and

K2Xi(n)

u=Ts

fi

u+

1

2

XNj= 1

fi

xj

K1Xj(n)

u

" #x1 = ^x1(n) +K1X1(n)=2

.

.

.

xN = ^xN(n) +K1XN(n)=2

49

and

K3Xi(n)u

=Tsfi

u+

1

2XNj= 1

fi

xj

K2Xj(n)u

" #x1 = ^x1(n) +K2X1(n)=2

.

.

.

xN = ^xN(n) +K2XN(n)=2

50

and

K4Xi(n)

u=Ts

fi

u+

1

2

XNj=1

fi

xj

K3Xj(n)

u

" #x1 = ^x1(n) +K3X1(n)

.

.

.

xN = ^xN(n) +K3XN(n)

51


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Thus, necessary derivatives for parameter estimation can be

obtained by using the RungeKutta model of the plant.

The simulation results

In the simulations, it is assumed that both process and mea-

surement noises are independent of each other and Gaussianwith zero-mean, and that covariance matrices Q and R are

diagonal matrices given by

Q =s2wI and R=s2vI 52

where I is the identity matrix and s2w and s2v are the variances

of the process and measurement noises, respectively.

The three-tank system

The first system on which the proposed scheme has been

tested is the three-tank liquid system as sketched in Figure 3.

The dynamics of the system can be expressed by a set of

differential equations (Iplikci, 2010)

_y1(t) =1

Au1(t) Q13(t)

_y2(t) =1

Au2(t) +Q32(t) Q20(t)

_y3(t) =1

AQ13(t) Q32(t) 53

where

Q13(t) = az13Snsgn(y1(t) y3(t))ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi2gjy1(t) y3(t)j

pQ20(t) = az20Sn ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2gy2(t)pQ32(t) = az32Snsgn(y3(t) y2(t)) ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffiffi2gjy3(t) y2(t)jp

For the system, ui(t) is the supply flow rate of pumpi as the

ith input and yi(t) is the liquid level of tanki as the ith output.

Explanations and values for the parameters appearing in the

system equations are given in Table 1.

In this work, the aim is to control the liquid levels of tank1and tank2 by manipulating flow rate of pump1 and pump2. In

the simulations, the sampling period is selected as Ts=1.0 s

and magnitudes of the control signals are allowed to be altered

between u1min =u2max= 0 m3/s and u1min =u2max=10

4 m3/s.

Furthermore, standard deviations of measurement and process

noises are sv=0.003 and sw=0.0001, respectively. Moreover,

because of the assumption that states of the system are not

available for measurement, they are estimated by the Runge

Kutta-based EKF block of the proposed structure.Figure 4 shows the simulation results for staircase reference

inputs. From the figure, it is seen that the proposed controller

carries out the control task very successfully: it provides very

small transient and steady-state tracking error despite the exis-

tence of both measurement and process noises. Moreover, it

effectively tolerates the interactions between the tanks, which

can be observed well during the periods when the level of a

tank is changing while the other one is remaining constant.

For better visualization of the performance of the pro-

posed controller, the reference input of tank2 is changed sinu-

soidally, while the reference input of tank1 is kept constant at

0.2 m, as seen in Figure 5. As can be seen from the figure,

even though the reference input for tank1 is constant, flowrate of pump1 is changing sinusoidally in order to compensate

for the interaction between the tanks.

Finally, in order to show the effectiveness of the online

parameter estimation capability of the proposed controller,

the reference inputs are set to 0.25 and 0.2 m for tank1 and

tank2, respectively, while varying the outflow parameter az13sinusoidally between 0.2 and 0.8. It is observed from Figure 6

that correct values of the parameter are estimated accurately

in a short period and then maintained in the long run.

3.2 The Van de Vusse chemical reaction

The second process on which the proposed structure has beentested is the Van de Vusse chemical reaction. It is a non-

isothermal process affected by thermal effect, and the result-

ing open-loop system shows strictly non-minimum-phase

behaviour. In this process, the following series/parallel reac-

tions take place

A!k1 B!k2 C2A!k3 D

54

The mass and energy balances describing the dynamics of the

process are given by

pump1 pump2

A

1y 2y3y

1u 2u

nSnS

DC

tank1 tank3 tank2

13az 32az10az 30az 20az

DC

Figure 3 The three-tank liquid level control system.

Table 1 The system parameters

Parameter description Value

az13 : outflow coefficient between tank1 and tank3 0.52

az32 : outflow coefficient between tank3 and tank2 0.55

az10 : outflow coefficient from tank1 to reservoir 0.26

az20

:

outflow coefficient from tank2

to reservoir 0.28az30 : outflow coefficient from tank3 to reservoir 0.45

A: cross section of the cylinders 0.0154 [m2]

Sn: section of connection pipe n 53105 [m2]

g: gravitation coefficient 9.81 [m/s2]

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_CA(t) =F

V(CA0

CA(t))

k10e

E1TCA

k30e

E3TC2A

_CB(t) = FVCB(t) +k10e

E1TCA k20e

E2TCB

_T(t) =1

rCpk10e

E1TCA( DH1) +k20e

E2TCB( DH2)

+k30eE3TC2A( DH3) +

F

V(T0 T(t)) + Q

rCp

55

where CA and CB are the molar concentrations of A and B,

respectively, T is the temperature of the reactor, FV

is the dilu-

tion rate, r is the density of the reacting mixture, CP is the

heat capacity, the DHi terms are the heat of the reaction, V is

the volume of the reactor and Q is the rate of heat added or

removed per unit volume (Niemiec, 2003). Nominal values of

the system parameters are given in Table 2.

For this process, the aim is to control the molar concentra-

tion of B (CB) and the temperature of the reactor (T) by

manipulating the dilution rate (FV

) and the rate of heat added

or removed per unit volume (Q)

In the simulations, the sampling period is selected as

Ts=0.01 h and magnitudes of the control signals are allowed

to be altered between [0,500] h21

and [1000,0] kJ/lh, respec-

tively. Moreover, the standard deviations of measurement

and process noises are sv =0.01 and sw =0.001, respectively.

Since the states of the system are assumed to be unavailable

for measurement, they are estimated by the RungeKutta-

based EKF block of the proposed structure.

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

time (sec.)

h1ref,

h1

h1ref

h1

0 200 400 600 800 1000 1200 1400 1600 18000

0.1

0.2

0.3

0.4

time (sec.)

h2ref,

h2

h2ref

h2

0 200 400 600 800 1000 1200 1400 1600 18000

0.2

0.4

0.6

0.8

1x 10

-4

time (sec.)

q1

0 200 400 600 800 1000 1200 1400 1600 18000

0.2

0.4

0.6

0.8

1x 10

-4

time (sec.)

q2

Figure 4 Inputs and outputs of the three-tank system for staircase reference inputs.


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For staircase reference inputs, the simulation results can

be seen in Figure 7. It is observed from the figure that the

proposed controller provides very small tracking errors except

for the short periods during which the reference input for the

reactor temperature is changed abruptly. During these peri-

ods, CB inevitably deviates from its reference value for a while

and then settles back to it. This is attributed to the strong

dependence ofCB on the reactor temperature.

Yet, it can be said that the proposed controller can effec-

tively tolerate the interactions between the state variables of

the process. As a matter of fact, as seen in Figure 8, the con-

troller is able to keep the temperature very close to the con-

stant reference input even though the reference input for CBis changing sinusoidally.

As has been done for the three-tank case, one parameter

(CA0 ) of the chemical reaction is sinusoidally changed around

its nominal value in order to test the adaptation capability of

the proposed structure. It can be observed from Figure 9 that

the actual parameter value is estimated very rapidly and accu-

rately by the online parameter estimation block and that the

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

0.1

0.2

0.3

0.4

time (sec.)

h1ref,

h1

h1ref

h1

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

0.1

0.2

0.3

0.4

time (sec.)

h2ref,

h2

h2ref

h2

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

0.2

0.4

0.6

0.8

1x 10

-4

time (sec.)

q1

0 200 400 600 800 1000 1200 1400 1600 1800 2000 22000

0.2

0.4

0.6

0.8

1x 10

-4

time (sec.)

q2

Figure 5 Inputs and outputs of the three-tank system for constant and sinusoidal reference inputs.

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0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

time (sec.)

h1ref,h1

h1ref

h1

0 100 200 300 400 500 600 700 800 900 10000

0.2

0.4

time (sec.)

h2ref,h2

h2ref

h2

0 100 200 300 400 500 600 700 800 900 10000

0.5

1x 10

-4

time (sec.)

q1

0 100 200 300 400 500 600 700 800 900 10000

0.5

1x 10

-4

time (sec.)

q2

0 100 200 300 400 500 600 700 800 900 1000

0.20.4

0.6

0.81

1.2

time (sec.)

az13

act. az13

est. az13

Figure 6 Inputs and outputs of the three-tank system for varying outflow parameter (az13) and its estimation

Table 2 Nominal parameters of the chemical reaction

k10 = 1:28731012 [h21] k20 = 1:287310

12 [h21] k30 = 9:0433109 [h21l/mol]

E1 = 9758:3 [K] E2 = 9758:3 [K] E3 = 8560:0 [K]DH1 = 4:2 [kJ/mol] DH2 = 11 [kJ/mol] DH3 = 41:85 [kJ/mol]CA0 = 5:0 [mol/l] T0 = 403:15 [K] r= 0:9342 [kg/l]Cp = 3:01 [kJ/kgK] V = 10:0 [l]


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controller can successfully tolerate the variations in the para-

meter of the process.

Computation times

Another important issue concerning the proposed structure is

the applicability to real-time systems. In order to have a basis

for the applicability of the proposed controller to the real-

time systems, the maximum response times of the proposed

controller have been calculated. For that purpose, the method

has been implemented in MatLab (version 7.4.0.287) on a PC

(with Intel(R) Core(TM)2 CPU T7200 running at 2.0 GHz

and 1024 MB RAM) without optimizing the codes, i.e. the

formulations given for each operation such as K-step ahead

predictions, Jacobian calculations, etc., have been strictly fol-

lowed. Then, the simulations have been carried out for the

staircase reference inputs, which are discussed in the previous

subsections. For each system under investigation, computa-

tion times of the operations and the resulting total response

times have been recorded during every sampling period. The

records corresponding to the maximum total response times

for the controller are tabulated in Table 3.

As can be seen from the table, the maximum total response

times of the proposed controller for both systems are less than

6 ms, which is much less than the sampling periods of the

investigated systems. Based on the results given in the table, it

0 5 10 15 20 25 30 35 40 45 500.6

0.8

1

1.2

time (hour)

CBref,CB

CBref

CB

0 5 10 15 20 25 30 35 40 45 50405.5

406

406.5

407

407.5

408

time (hour)

Tref,T

Tref T

0 5 10 15 20 25 30 35 40 45 500

100

200

300

400

500

time (hour)

F/V

0 5 10 15 20 25 30 35 40 45 50-1000

-800

-600

-400

-200

0

time (hour)

Q

Figure 7 Inputs and outputs of the chemical reaction for staircase reference inputs.

Iplikci 177

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can be stated that the proposed controller can be used in real-

time applications.

Conclusions

This paper proposes a structure that introduces the use of the

fourth-order RungeKutta numerical integration tool for con-

structing the so-called RungeKutta models for non-linear

processes and consequently its utilization in the GPC frame-

work. In the proposed structure, the role of the RungeKutta

model of a non-linear process is fourfold: (a) it accounts for

future predictions of the process in response to the candidate

control vector; (b) it is used to extract the gradient informa-tion required for the Jacobian calculation; (c) it is used in the

EKF framework in order to estimate the unmeasurable states

of the system; (d) it is used for online estimation of a time-

varying parameter of the process.

The proposed structure has been tested on two different

non-linear MIMO processes by simulations under various sce-

narios. The simulation results have revealed that the proposed

structure is capable of (a) providing very small transient and

steady-state tracking errors for constant reference inputs and

system parameters; (b) adaptation to the changes in the

0 5 10 15 20 25 30 35 40 45 500.6

0.8

1

1.2

time (hour)

CBref,CB

CBref

CB

0 5 10 15 20 25 30 35 40 45 50405.5

406

406.5

407

407.5

408

time (hour)

Tref,T

Tref T

0 5 10 15 20 25 30 35 40 45 500

100

200

300

400

500

time (hour)

F/V

0 5 10 15 20 25 30 35 40 45 50-1000

-800

-600

-400

-200

0

time (hour)

Q

Figure 8 Inputs and outputs of the chemical reaction for constant and sinusoidal reference inputs.


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reference inputs; (c) compensation of the interactions between

the state variables of the processes; (d) online estimation of

the time-varying system parameter and (e) tolerating the

effects of the time-varying system parameter.

As a result, the proposed structure, containing several

novelties, can be used not only for control but also for state

estimation and online parameter estimation purposes.

Funding

This research received no specific grant from any funding

agency in the public, commercial, or not-for-profit sectors.

0 5 10 15 20 25 30 35 40 45 50

0.6

0.8

1

1.2

time (hour)

CBref,CB

CBref

CB

0 5 10 15 20 25 30 35 40 45 50

406

407

408

time (hour)

Tref,T

Tref T

0 5 10 15 20 25 30 35 40 45 500

200

400

time (hour)

F/V

0 5 10 15 20 25 30 35 40 45 50-1000

-500

0

time (hour)

Q

0 5 10 15 20 25 30 35 40 45 50

4.5

5

5.5

6

6.5

time (hour)

CA

0

act. CA

0

est. CA

0

Figure 9 Inputs and outputs of the chemical reaction for varying concentration parameter ( CA0 ) and its estimation.

Table 3 Computation times for the proposed controller

Computation time [ms.]

Operation Three-tank Chemical reaction

State estimation 0.49 0.39

K-step ahead prediction 0.68 0.33

Jacobian calculation 2.23 2.57

Parameter estimation 1.77 0.92

Miscellaneous 0.12 0.12

Total 5.29 4.33

Iplikci 179

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