HUNGARIAN JOURNAL 
OF INDUSTRIAL CHEMISTRY 

VESZPRÉM 
Vol. 33(1-2). pp. 89-96  (2005) 

ROBUST MODEL PREDICTIVE CONTROL WITH STATE ESTIMATION  FOR 
AN INDUSTRIAL PRESSURIZER SYSTEM 

P. TAMÁS, I. VARGA, G. SZEDERKÉNYI  and  J. BOKOR 

Systems and Control Laboratory,  
Computer and Automation Research Institute, HAS, 

 H-1518 Budapest, P.O. Box 63, HUNGARY 
 

 Robust model predictive control of an industrial pressurizer is presented in this paper. The physical model of 
the system is based on first engineering principles and the model parameters have been previously identified 
from measured data. To satisfy the hard constraints on the state variables and the input even in the presence 
of disturbances, the so-called single policy robust model predictive control method is applied. The maximal 
admissible level set, the disturbance invariant set and the terminal sets are determined for the system. 
Simulation results show that the proposed controller satisfies all the requirements and shows good time-
domain behavior. 

Keywords: process control, robust model predictive control, constrained control 

Introduction 

Recently, there has been a growing need for automation 
of increasingly complex plants in different branches of 
industry. Fortunately, the improving quantity and 
quality of measurements and actuators allows us to 
apply an increasing variety of techniques in systems and 
control theory developed in the last few decades. 
 
This paper presents a robust model predictive control 
for an industrial pressurizer system used mainly for 
pressure control in a nuclear power plant. In [9] an 
advanced dynamic inversion-based pressure controller 
has been designed for the systems of this type. Although 
this controller performs very well in practice, our aim is 
to further develop control performance. For this, a 
robust model predictive control approach is proposed in 
this paper that is able to handle the strict input and state 
constraints even if disturbance affects on the system. 

The dynamic model of the system 

System description 

The system discussed is a pressurized water reactor, 
which means that in the primary circuit high pressure 
ensures that the coolant is not boiling. The task of the 
pressurizer is to keep the pressure within a predefined 
range. From a modeling point of view, the pressurizer is 
a vertical tank and inside this tank there is hot water at a 
temperature of about 326°C and steam above. If the 
primary circuit pressure decreases, water might start to 
boil. In order to prevent this, electric heaters switch on 
automatically in the pressurizer. Due to the heating 
there will be intense boiling, more steam will be 
generated and this leads to a pressure increase. If the 
increasing pressure in the pressurizer reaches a certain 
limit, firstly the heaters are turned off and then cold 
water is injected into the tank (if needed) to reduce the 
pressure down to the predefined range [7]. The heating 
power of the electric heaters can be set continuously. 
The measured outputs of the system are the pressure in 
the pressurizer and the temperature of the tank wall. The 



 90

controlled output is the pressure in the tank. The 
simplified flowsheet of the pressurizer is shown in Fig. 
1. 

 

YP10

s

kg
m

s
kg

m

.constM =

1χ 4χ3χ2χ

[ ]KT I

[ ]KT

 
Figure 1: simplified flowsheet of the pressurizer 

The physical model of the plant 

Modeling of industrial systems  depends heavily on the 
modelling goal. Most of the commercially available 
dynamic models are implemented in  simulators (see 
e.g. [1]) and are used for equipment design  and 
retrofitting purposes. The models used in this area are 
typically in the form of partial differential equations 
 that are discretized in space to have a lumped version. 
This way a high dimensional (with 10-100 state  
variables) complicated  dynamic model is obtained that 
is unnecessarily complex for control  applications. 
 
Instead, a simplified lumped dynamic model is 
constructed for control design purposes based on first 
engineering principles [2] that captures the most 
important dynamics of the  tank. For this purpose, the 
following assumptions were used: 
 
1. There are two perfectly stirred balance volumes, one 

for the water and another for the wall. 
2. There is a single component in each of the balance 

volumes (water and metal, respectively). 
3. Constant overall mass in both balance volumes. 
4. Constant physico-chemical properties. 
5. Vapour-liquid equilibrium in the tank. 
 
The simplified model consists of two energy balances: 
one for the water and  and another  one for the wall of 
the tank as balance volumes. 
 
Water energy balance 
 

( )= − + − +p I p W W HE
dU

c mT c mT K T T W
dt

⋅ χ  (1) 

 
Wall energy balance 

 

  ( - ) -  =W W W lo
dU

K T T W
dt ss

 (2) 

 
 

The following constitutive equations, describing the 
relationship between  the internal energies and the 
corresponding temperatures, complete the model. 
 

 = 

= p

W pW

U c MT

U C TW
 (3-4)  

 
The variables and parameters of the above model and 
their units of measure are the following 
 

T water temperature °C 
WT  tank wall temperature °C 

pc  specific heat of water 
J

kg°C
 

U internal energy of water J 
WU  internal energy of the wall J 

m 
mass flow rate of water 

kg
s

 

IT  inlet water temperature °C 
M mass of water Kg 

pWC  heat capacity of the wall 
J

°C
 

HEW  total heating power of one electric 
heater 

W 

χ  portion of total heating power 
turned on 

- 

WK  wall heat transfer coefficient 
W
°C

 

lossW  heat loss of the system W 
 
The manipulable input to the system is the external 
heating, all the other input variables are regarded as 
disturbances. Then we can list the disturbances with 
physical meaning as  follows. 
 
• Cold water infiltration.  This effect is taken into 

account with the in-convection term  in the 
water energy conservation balance (1), where the in- 
and outlet mass flowrate m is controlled to be equal 
(but might change in time)  and the inlet temperature 

 can also be time-varying.   

p Ic mT

IT
 
• Energy loss towards the environment. This effect is 

modelled as a loss term  in the wall energy 
balance   (2). 

lossW

 
 
 
The pressure of saturated vapor in the  gas phase of the 
tank depends strongly on the water temperature  in an 
exponential (nonlinear way). The experimental 
measured data found in the literature  [7] have been 

 



 91

used to create an approximate  analytic function to 
describe the dependence. The function has the form 
 

 

( )

2 3
0 1 2 3

( )
100

( )

= =

= + + +

Te
p h T

T c c T c T c T

ϕ

ϕ
 (5) 

 
For the parameters of ϕ , the following values were 
obtained 
 

 
1

0 1
5

2 3

6.5358 10 4.8902 10

9.2658 10 7.6835 10

− 2

8

−

− −

= ⋅ = ⋅

= − ⋅ = ⋅

c c

c c
 (6) 

 
 
The validity range of the model is the usual operating 
domain of the pressurizer, i.e. 315°C ≤ T ≤ 350°C. In 
pressure terms, this means 105.65 bar ≤ p  ≤ 137.09 bar. 
 
 

State space description 
 
Based on eqs (1)-(2) and (3)-(4) we can write the system 
model in the following standard state-space form 
 
 = + +& c c cx A x B u E d  (7) 
 
where the state vector [= TWx T T ] , the manipulable 
input u  is directly proportional to the heating power, 
and the disturbance input vector [ ]= TI lossd T W . 
Furthermore, the matrices in (7) are the following 
 

0
, 1

00

W W

p p
c

W W

pW pW

HE

pc c

pW

K Km
M c M c M

A
K K

C C

mW
M

c MB E

C

⎡ ⎤
− −⎢ ⎥
⎢ ⎥= ⎢ ⎥

−⎢ ⎥
⎢⎣

⎡ ⎤
⎡ ⎤ ⎢ ⎥
⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥⎣ ⎦

⎣ ⎦

⎥⎦  (8-9) 

 
The real physical measured variable in the system is the 
pressure. Since (according to our assumptions) the 
temperature in the tank is a monotonous and invertible 
function of the pressure, we can write a linear output 
equation in the form 
 
 [ ]1 0y = x  (10) 
 
 
This means that we can express the performance 
requirements (bounds) for the pressure in terms of the 
temperature in the tank.  

The unknown parameters of the continuous-time 
model have been estimated from input-output 
measurement data. The model structure together with 

the estimated parameter values have been successfully 
validated. The detailed system identification procedure 
is described in [11]. 

For the controller design, it is convenient to center 
the state, input and disturbance variables as follows: 

 
 , ,x x x u u u d d d∗ ∗ ∗= − = − = −  (11) 
 

where x∗  is the required steady state to be kept by the 
controller, d ∗  is the nominal (mean) value of the 
disturbances and u∗  is the constant input necessary to 
keep the prescribed steady state x∗ . Using the centered 
coordinates, the system model (7)-(9) can be rewritten 
as 
 

 c c cx A x B u E d= + +&  (12) 
 
For the controller design, we need a discrete-time state 
space model of the system (12). The discretization was 
performed assuming zero order hold on the input with a 
sampling interval of 10s. The centered discrete-time 
model is given by the equations 
 
 1k k k kx Ax Bu w+ = + +  (13) 
 
where the effect of the disturbance term  is 
expressed in discrete-time by the state disturbance . 

cE d

kw
 
 

Robust model predictive control design 
 

Control problem formulation 
 
The aim of the control is the robust stabilization of a 
prescribed operating point in case of additive 
disturbance, while the state and the control are subject 
to hard constraints. To solve this problem we apply the 
single policy robust model predictive control proposed 
by Langson et al. in [5]. Since this method requires the 
knowledge of the entire state, in our case it has to be 
completed with an appropriate state estimator. The 
control design procedure will then consist of three steps: 
first, assuming that the full state is available for 
measurement and no disturbance affects on the system, 
a nominal MPC controller is designed. Then the 
robustification of the nominal controller comes making 
the nominal MPC applicable in the presence of external 
disturbances. The last step is the design of a state 
estimator and its integration into the MPC control 
framework. 
 

Before starting the procedure the following 
assumptions have to be made: 
 
• the disturbance is bounded and there exists a convex 

polytope W containing the origin in its interior s.t. 
kw W k∈ ∀   

• the constraints on the state and the control input are 
convex, i.e. there are given convex, polytopic sets X 

 



 92

and U containing the origin in their interior s.t. 
, ku U∈ kx X∈  have to be hold . For simplicity 

we moreover assume that U is rectangular i.e. 
k∀

1 1 2 2
[ , ] [ , ], [ , ]

p pL L L L L L
U u u u u u u= − × − × × −L  . 

 
Nominal MPC 

 
Following the steps of control design procedure in [5] 
we have to first formulate and solve the model 
predictive control problem for the disturbance free case. 
This means the determination of an admissible receding 
horizon policy ( )kxµ , which steers the centered system 
from initial state 0x X∈  to the origin. The solution to 
this problem can be derived from the result of the 
following optimization problem: 
 

 

1

0

0 1 0 1

1 0 0

arg min ( )

( ) ( ( ), ) ,

[ , , ], [ , , ]
,

,

v N
N

T T
N N i i i

i

N N

i i i

i i

k N f

v V v

V v V x v v x Qx v Rv

v v v x x x
x Ax Bv x x
x X v U
x X

∗

−

=

− −

+

+

=

= = +

= =

= + =

∈ ∈

∈

∑
K K

i

 (14) 

 
where N is the length of the horizon and fX X⊂  is a 
terminal set having the following properties: there 
existsan admissible static state feedback control 

, which keeps the system trajectories in ( )u x Fκ= = x

fX  i.e.  
 
 for all ( ) , ( )f fx X x U Ax B x Xκ κ∈ ∈ + ∈  (15) 
 
and asymptotically stabilizes the system i.e. 
 

1 0lim 0 if ( ) and k k k k k fx x Ax B x x Xκ→∞ += = + ∈  
 (16) 
  
In possession of the optimal control input vector v∗  we 
can formulate the single policy MPC controller in the 
following way:  
 

 nom
if

( )
if

i
i

i

v i
x

N
Fx i

µ
∗⎧ ≤⎪

= ⎨
>⎪⎩ N

 (17) 

 
i.e by means of single policy approach we determine v∗   
only in the beginning (and later only if the prescribed 
operating point changes),  and after depleting the entire 
control sequence, we switch to the linear feedback Fx 
(dual-mode control). 
 
It can be easily proved [5], [6] that the control policy 
(17) asymptotically stabilizes the plant in the 
disturbance free case and both the state and the input 
will satisfy the constraints. The Lyapunov function for 

the closed loop dynamics can be derived from the 
quadratic cost . NV
 
To implement the formulated MPC algorithm on a 
particular system we need to determine the feedback 
gain F and the associated terminal set fX . It is 
straightforward to choose F as an unconstrained LQ 
controller minimizing the infinite horizon cost function 
defined as:  
 

 
0

1

( ) ( , )

, ,
, 0

T T
i i i

i

k k k k k

V v V x v x Q x v R v

v Fx x Ax Bv x x
Q R

∞

∞ ∞ ∞ ∞
=

+

∞ ∞

= = +

= = +
>

∑

0 0

i

=

k

 (18) 

 
The solution F and the quadratic Lyapunov function 

 of the closed loop dynamics ( ) TW x x Px=

1 ( )kx A BF x+ = +  can be obtained as a solution of a 
discrete algebraic Ricatti equation:  
 

  (19) 
1( ) ( ),

( )( )( )

T T

T T T T T

F B PB R B PA

P A PA A PB B PB R A PB Q

−
∞

∞ ∞

= +

= − + +
 
Since the terminal set fX  has to be invariant for the 

dynamics 1 ( )k kx A BF x+ = +  it can be constructed from 
an appropriate level set of W(x). Let FX  be the 
maximal level set, which satisfies the input constraints, 
i.e. 

,

{ | }

max , { | max }T
i

T
F

T
i Lx x Px

X x x Px

f x u iγ γ

γ

γ γ γ

∗

∗
∈Γ ≤

= ≤

= Γ = ≤ ∀
 

 (20) 
 
where Tif  is the ith row of F. Notice that 

 is a support function of the set 
,

( ) max T Ti x x Pxh f f xγ≤= i

{ | }Tx x Px γ≤ , which can be calculated as 
1( ) Ti ih f f P fγ
−= i   ([4],[3]). Consequently in single 

input case 

 
2

1
, TL

T
u

F f
f P f

γ ∗
−

= =  (21) 

Since FX  may be larger than X, let f FX X X= ∩ . 
(The numerical calculation of the intersection can be 
greatly simplified if the polytopic approximation of X  
and FX  is used.) 
 

Robust MPC 
 
The next step of the controller design is the 
"robustification" of the nominal control policy. This can 
be performed by tightening the constraints of the 
nominal MPC and completing the nominal control input  

nom ( )ixµ  with an appropriate error feedback term, i.e. 
more precisely: 

 



 93

 
 nom( ) ( ) ( )i i i ix x K x xµ µ= + −  (22) 
 
where ix  is the nominal state value – calculated by 
iterating 1 nom ( )k k kx Ax B xµ+ = + i, x  is the true state 
(measured / estimated),   and K  is a stabilizing feedback 
for the disturbance-free dynamics 1k k kx Ax Bu+ = + .  (It 
is possible to choose K F= . In order to formalize the 
new, tighter constraints for the robust MPC problem, we 
have to determine the following disturbance invariant 
set:  
 

 
0
( )i

i
Z A BK W

∞

=
= +∑  (23) 

 
Because of the infinite summation the equation above 
can not be applied directly. There are two possibilities: 
we can use an approximation for Z ([8]), or we can 
apply (23) till the difference between two consecutive 
sets becomes smaller than the numerical accuracy of the 
computing software. The first approach is 
mathematically precise, but we used the second one, 
since it is easier to implement and the convergence of 
(23) is fast enough to make this procedure practically 
applicable. Using Z  the stringent sets of constraints can 
be calculated as follows: 
 
 , f fX X U U KZ X X= ∼ Ζ = ∼ , = ∼ Z

}

 (24) 
 
where ~ denotes the Pontryagin difference of two sets, 
defined as: 
 
 ~ { |A B x x B A= + ⊂  (25) 
 
At this point we can formulate the robust MPC 
procedure: 
 

 

1

0

0 1 0 1

1 0 0

arg min ( )

( ) ( ( ), ) ,

[ , , ], [ , , ]
,

,

v N
N

T T
N N i i i

i

N N

i i i

i i

k N f

v V v

V v V x v v x Qx v Rv

v v v x x x
x Ax Bv x x

x X v U

x X

∗

−

=

− −

+

+

=

= = +

= =

= + =

∈ ∈

∈

∑
K K

i

 (26) 

i.e. we follow the same procedure as (14), but – instead 
of , , fX U X  – we use the tightened constraint sets 

, , fX U X .  According to (22) the control policy is 
defined as: 
 

 
( ) if

( )
( ) if

i i i
i

i i i

v K x x i N
x

Fx K x x i N
µ

∗⎧ + − ≤⎪
= ⎨

+ − >⎪⎩
 (27) 

 
 
 
 

State estimation 
 
As we have mentioned in the first section  the state 2x  
of the pressurizer can not be measured directly. For this, 
we apply discrete-time state estimator to approximate it 
on-line from the input u  and the measured output 

1y x= . The robust MPC controller will then work with 
this estimated state.  
 
The estimator applied is given in the following well-
known form: 
 

 
1| 1 1| 1 1|

1| |

ˆ ˆ ( )
ˆ ˆ
k k k k e k k k

k k k k k

x x K y Cx

x Ax Bu
+ + + + +

+

= + −

= +
 (28) 

 
where |k kx  denotes the estimated state at time instant k. 
Substituting (13) into (28) the following error dynamics 
can be derived: 
 

 
1 1 1| 1

|

ˆ

( )( ) (

( ) ( )

k k k k

e k k k e

e k e k

e x x

) kA K CA x x I K C w

A K CA e I K C w

+ + + += −

= − − + −

= − + −

 (29) 

 
where eK  is chosen so that the matrix eA K CA−  will 
be stable. Before applying the estimator we have to 
examine the effect of the estimation error on the 
stability of the controlled system and on the prescribed 
constraints. We examine the system behaviour after the 
disappearence of the initial transients of (29), i.e. we 
assume that the estimation error is caused only by the 
disturbance , and not the initial difference between 
the estimated and the true states. By iterating (29) we 
can easily see that after some steps 

kw

|ˆk k k ex x Z k∈ + ∀ ,  
where eZ  is a disturbance invariant set constructed in 
the following way: 
 

 
0
( ) (ie e e

i
)Z A K CA I K C W

∞

=
= − −∑  (30) 

 
Since by (13) 1k k kx Ax Bu W+ ∈ + +  holds and eZ  is 
symmetric to the origin, for the estimated state a 
following relation can be derived: 
 

1| 1 | |

1| 1 |

ˆ ˆ ˆ

ˆ ˆ ,

k k k k k e e k k k e

k k k k k k k e

x Ax Bu W AZ Z Ax Bu W

x Ax Bu w w W

+ +

+ +

∈ + + + + = + +

⇓
= + + ∈

 (31) 
  
Thus, if we perform the same controller synthesis as 
before with  instead of W  we get a control policy eW

ˆ( )kxµ  which guarantees the stability of (31) while 
keeping the state ˆkx  and the input in the predefined 
range. Since the difference between the true and the 

 



 94

estimated state is considered in  the real state will 
also satisfy these constraints. Since  is generally 
larger than W  the sets 

eW

eW
, , fX U X  constructed from  

prescribe much tighter constraints than those which are 
constructed from W . 

eW

 
 

Simulation results 
 

To demonstrate and examine the performance of the 
controller designed above an identified model of the 
pressurizer was used [10]. After discretization the 
following state space model was obtained: 
 

0.6651 0. 0.1035
,

0.0355 0. 0.0024
A B

⎡ ⎤ ⎡ ⎤
= =⎢ ⎥ ⎢ ⎥
⎣ ⎦ ⎣ ⎦

91

21

=

3341
9645

 (32) 

 
For the simulation two reference values were chosen: 
 
   (33) 327.166, 326.166a br ry y= =
 
The corresponding operating points are as follows: 
 

  (34) 

327.1660
, 1.71

326.7760

326.1660
, 1.71

325.7760

a a
ss ss

b b
ss ss

x u

x u

⎡ ⎤
= =⎢ ⎥
⎣ ⎦
⎡ ⎤

= =⎢ ⎥
⎣ ⎦

 
The LQ controller was designed by using the following 
weighting matrices: 
 
  (35) 210 , 20Q I R∞ ∞=
 
The resulted feedback gain is [ ]0.1439 0.8392F = . 
The following constraints for state and input were 
chosen by considering the physical limitations of the 
system: 
 
  (36) 1 21.5 1.5, 3 3, 1.71 1.71x x u− ≤ ≤ − ≤ ≤ − ≤ ≤
 
The cause why the constraint on the control input was 
constructed in the form above is the following: in the 
true system the control input has to be between 0 and 4, 
which is equivalent to constraints 4ss su u u− ≤ ≤ − s  for 
the centered model. But it is more convenient to handle 
constraints, which are independent from the operating 
point and symmetric to the origin, so we restrict the 
bounds to [ ]1.71 1.71−  where 1.71 min( , )a bss ssu u≤  
 
By examining the system behavior, it was seen that the 
difference between the nominal  and the true d ∗ d  
disturbance is at most 15%, which means that w in the 
centered, discrete model is inside the rectangle defined 
as: 
 
 [ ] [ ]0.05 0.05 0.005 0.005w W∈ = − × −  (37) 

We used Kalman filter as a state estimator, which was 
designed according to the measurement noise 
conditions. The obtained filter gain was 

[ ]0.7712 0.5982eK = . 
 
Setting the stabilizing K controller equal to F the sets 
needed for the MPC algorithm can be calculated. Figure 
4. 5. shows the results in the case when full-state 
measurement is assumed and Figure 3. 4. shows the 
obtained constraint sets in the case of state estimation. It 
can be seen that the constraints to be satisfied by the 
input and states are much more tighter if state estimator 
is applied. 
 
In the simulation the horizon was N=50, the system 
started from  and the weighting 
matrices in the MPC optimization were chosen to be 

[0 327.5 327
T

x = ]

0i
 
 2 ,iQ i I R= ⋅ =

s

 (38) 
 
The system had to track  first, and this reference was 

changed to   at time 

a
ry

b
ry 24000t = . To illustrate the 

robustness of the controllers designed the maximal 
15%±  persistent disturbance was added to the original 

plant. Figure 6 shows the factors by which the nominal 
disturbance was multiplied.  
 
The simulation results in the cases of full state 
measurement and state estimation the can be found in 
Figures 7. and 9. Figure 8. shows the state estimation 
error. 
 
It can be seen that in all cases the output remains in the 
1.5-wide neighborhood of the prescribed value while the 
control input also satisfies the constraints. It is also seen 
that from reference tracking point of view there is no 
significant difference between the cases of full state 
measurement and state estimation. although the 
controller using estimated states has to satisfy more 
stringent constraints to achieve the same result. 
Examining the settling performance it can be stated that  
the overshoot is negligible and the setting time is 
acceptable small. The time-consuming optimization 
procedure was executed only twice: first, in the 
beginning and later at 24000t s= , when the change of 
reference took place.  
 

 
 

Conclusions and further work 
 
A single policy robust model predictive control was 
successfully applied to a pressurizer model. It was 
shown that the states and the stabilizing control input 
designed by this approach remain in the given range 
even if additive disturbance is present. By using single 
policy approach the computation time needed for the 
control input has been drastically reduced. 
 
 

 



 95

 
Acknowledgements 

 
This research was partially supported by the grants no. 
OTKA T042710 and F046223.  
 
 

 
Figure 2 : Constraint sets in the case of state estimation 

, , ,F eX Z X X  

 

 
Figure 3 : Constraint sets in the case of state estimation 

, , ,f fZ X X U  

 
 
 

 
Figure 4: Constraint set in the case of full state 

measurement , ,FX X X  

 
 
 

 
 

Figure 5 : Constraint set in the case of full state 
measurement , , ,f fZ X X U  

 
 

 



 96

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10
4

0

0.5

1

1.5

2

2.5
u

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10
4

325.5

326

326.5

327

327.5

328
y
yr

Figure 9 : Reference signal  (solid), system output  
(dashed) and control input in the case of state estimation 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10
4

0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25
d1
d2

 
Figure 6 : Disturbance scaling factors used in 

simulation. 

REFERENCES 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10
4

325.5

326

326.5

327

327.5

328
y
yr

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10
4

0

0.5

1

1.5

2

2.5
u

 

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4. I. KOLMANOVSKY AND E. G. GILBERT.: Theory and 
computation of dysturbance invariant sets for 
discrete-time linear systems. Mathematical 
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5. W. LANGSON, I CHRYSSOCHOOS, S. V. RAKOVIC 
AND D. Q. MAYNE: Robust model predictive control 
using tubes. Automatica, 40:125-133, 2004. 

Figure 7 : Reference signal (solid), system output  
(dashed) and control input in the case of full state 

measurement 

 

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10
4

−0.1

−0.05

0

0.05

0.1

0.15

0.2

0.25
x1−x1e
x2−x2e

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 10
4

325

325.5

326

326.5

327

327.5

328
x1e
x1
x2e
x2

 

6. D. Q. MAYNE, J. B. RAWLINGS, C. V. RAO AND P. O. 
M. SCOAKERT: Constrained model predictive 
control: Stability and optimality. Automatica, 
36(3):789-814, 2000. 

7. R. H. PERRY AND D. W. GREEN: Perry’s Chemical 
Engineers’s Handbook (7th edition). Mc Graw Hill, 
New York, 1999. 

8. S. V. RAKOVIC, E. C. KERRIGAN, K. KOURAMAS 
AND D. Q. MAYNE.: Approximationof the minimal 
robustly positively invariant set for discrete-time 
LTI systems with persistent state disturbances. 42th 
Conference on Decision and Control, 2003. 

9. Z. SZABÓ, P. GÁSPÁR AND J. BOKOR: Reference 
tracking of Wiener systems using dynamic 
inversion. In 2005 International Symposium on 
Intelligent Control, Limassol, Cyprus, pages on CD, 
paper ID: WeA06.5. 2005. 

10. I. VARGA, G. SZEDERKÉNYI, K. M. HANGOS AND J. 
BOKOR: Modelling and model identification of a 
pressurizer at the Paks Nuclear Power Plant. In 14th 
IFAC symposium on System Identification. 
(submitted), 2006. 
 

Figure 8 : True (solid) and estimated (dashed) states and 
estimation error. 

 

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