Implementation of Centralized MPC on the Quadruple-tank Process

with Guaranteeing Stability

Roza Ranjbar

, Lucien Etienne

1 a

, Eric Duviella

1 b

and Jos

e Mar

ıa Maestre

2 c

Institute Mines Telecom Lille Douai, Univ. Lille, F-59000, Lille, France

Dept. Ingenier

ıa de Sistemas y Autom

atica, Universidad de Sevilla, Sevilla, Spain

Keywords:

Model Predictive Control, Centralized Control, Control Benchmark, Stability.

Abstract:

This work presents an implementation of a stabilizing model predictive control applied to a nonlinear system.

In this work, the quadruple-tank system has been considered. For this process, a precise control benchmark

was available and worked on previously. To ensure the asymptotic stability of this nonlinear system, we made

a discretized linearized model and applied a centralized MPC controller with terminal cost constraint. The

effectiveness of the proposed strategy is illustrated by simulations.

1 INTRODUCTION

Optimal control design of systems subject to con-

straint is an important problem of control theory.

A powerful way of investigating this problem is to

use model predictive controllers (MPCs) which are

known to be popular in many ﬁelds of applications

(Qin and Badgwell, 2003). MPC uses a model of

the system dynamics for computing an optimal con-

trol action sequence therefore enhancing the compu-

tational requirements while achieving optimal perfor-

mance (Dua et al., 2006). It solves an open-loop con-

strained optimization problem at each time step, then

it executes only the ﬁrst control of this sequence. The

same procedure is repeated at next time steps (Seung

Cheol Jeong and PooGyeon Park, 2005).

One of the major beneﬁts of MPC over the other

controllers is that it can manage constraints on states,

inputs, and outputs. Thus, it allows a system to op-

erate closer to boundaries (Huang et al., 2017). In

addition, MPC has the ability of tracking a consistent

sequence of set points at the same time that it guaran-

tees that the constraints are satisﬁed at all times (Al-

varado et al., 2011).

MPC strategies have been considered for linear

and nonlinear systems, under a variety of communi-

cation schemes such as centralized MPC, decentral-

ized MPC, distributed MPC (Segovia et al., 2019;

https://orcid.org/0000-0003-0931-843X

https://orcid.org/0000-0002-1622-0994

https://orcid.org/0000-0002-6343-5445

Fele et al., 2017). In this paper, we propose a frame-

work for analyzing the implementation of a classical

centralized MPC to ensure the stability of a popu-

lar benchmark example of the quadruple-tank process

with nonlinear dynamics (Johansson, 2000). We will

also emphasize that an optimally controlled system is

not necessarily stable and the stability is not ascer-

tained by the use of a ﬁnite horizon optimal controller

(Kalman et al., 1960; Pannocchia, 2012; Scokaert and

Rawlings, 1998).

Related works: Previous researches have been

done to provide sufﬁcient conditions for the stability

of a MPC controller. Since (Mayne et al., 2000) indi-

cates that stability is an overriding necessity resulting

in varied proposals for a MPC and its formulations.

Later on, (Cueli and Bordons, 2008) studied a case

(both constrained and unconstrained) for deriving the

stability criterion that could be ensured under some

speciﬁc assumptions. Afterwards, (Maiworm et al.,

2015), by using a scenario tree, proved how to ensure

a reasonable level of stability in the performance of

the MPCs.

Contributions: Based on the approach of

(Scokaert and Rawlings, 1998) and by using conti-

nuity arguments, the main contribution of this paper

is to provide sufﬁcient conditions for the stability of

a nonlinear system comprised of four-tanks (as a rep-

resentative of a water network) controlled with a cen-

tralized MPC. Namely, we propose a framework for

proving asymptotic stability of a Lipschitz nonlinear

system using a discretized linearized model for the

MPC controller synthesis. Then we apply this result

Ranjbar, R., Etienne, L., Duviella, E. and Maestre, J.

Implementation of Centralized MPC on the Quadruple-tank Process with Guaranteeing Stability.

DOI: 10.5220/0009827700560062

In Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2020), pages 56-62

ISBN: 978-989-758-442-8

on the four-tank benchmark example. The proposed

stability outcomes can be applied to any other system

which satisﬁes the assumptions made in section 3.

Outline: Section 2 introduces the the dynami-

cal system under consideration as well as the control

scheme. In Section 3, the stability analysis is con-

ducted. Section 4 present the benchmark example.

The results are discussed in Section 5 while in Sec-

tion 6 some concluding remarks are presented.

2 PROBLEM STATEMENT

2.1 System Dynamics

Consider a nonlinear system

˙x = Ax + Bu + f (x, u), (1)

with (x,u) = (0,0) an equilibrium point such that f is

locally Lipschitz : ∀u ∈ U,x

∈ X

||Ax

+ f (x

,u) − Ax

− f (x

,u))|| ≤ L||x

− x

||,

where U ⊂ R

and D ⊂ R

are open set containing 0

with non empty interior.

We make the following assumptions

• A1: || f (x,u)|| ≤ γ(|x|)||x|| with γ(|x|) a class K

function,

• A2: (A,B) is stabilizable.

First we treat the nonlinear term f (x, u) as a per-

turbation/noise. Close to equilibrium assumption A1

quantiﬁes the fact that the nonlinear term can be ne-

glected. We have

˙x ≈ Ax + Bu. (2)

Given h > 0 a sampling period, we will introduce

the exact discretization of (A,B) by (A,B). Writ-

ing x

= x(kh) and u

= u(kh), we obtain a general

nth-order discrete-time linear state-space description

which takes the following form

k+1

= Ax

+ Bu

, (3)

where x

∈ R

. Assigning the x

as the initial con-

dition and u

∈ U ⊆ R

as the discrete time input,

x(k) ∈ X ⊆ R

is system’s state. U is the set of ad-

missible input and we assume it has a non empty in-

terior, while X is the set of admissible state. We also

assume that it has a non empty interior (Rawlings and

Mayne, 2009).

2.2 Model Predictive Controller

By applying a centralized MPC for the system (3) the

following optimization problem should be solved

min

i|k

}

k+H

−1

i=k

,{x

i|k

}

k+H

i=k

J({u

i|k

}

k+H

−1

i=k

,{x

i|k

}

k+H

i=k

), (4)

with {u

i|k

}

k+H

−1

i=k

, {u

k|k

k+1|k

,..., u

k+H

−1|k

} and

i|k

}

k+H

−1

i=k

, {x

k|k

k+1|k

,..., x

k+H

−1|k

}, with

J({u

i|k

}

k+H

−1

i=k

,{x

i|k

}

k+H

i=k

) =

k+H

−1

∑

i=k

i|k

+ u

i|k

+ x

k+H

Here Q is the weight associated to the states, R the

weight associated to the outputs, Q

is a terminal cost

on the state and H

is the prediction horizon. Q

chosen to be the solution to the discrete time Riccati

equation associated with (A,B,Q,R), i.e. the solution

= A

A − (A

B)(R + B

−1

A) + Q

subject to the following constraints

i+1|k

= Ax

i|k

+ Bu

i|k

,i ∈ {k,..., k + H

− 1}, (5a)

i|k

∈ U, i ∈ {k,...,k + H

− 1}, (5b)

j|k

∈ X , j ∈ {k, ...,k + H

}, (5c)

k|k

= x

. (5d)

Constraint (5a) shows the state equation presented in

(5); (5b) describes the feasible inputs and (5c) the

feasible states. Finally, the constraint (5d) represents

the system’s initial condition. In this centralized ap-

proach, only the ﬁrst input u

k|k

is applied to the sys-

tem (see (6)) and the others are being neglected ac-

cording to the receding-horizon philosophy (Richter

et al., 2009) (which the control is repeated in this phi-

losophy at every time-step and gives the information

of the new state). The following action will be imple-

mented at each time step

MPC

, u

k|k

. (6)

3 STABILITY ANALYSIS

Let us recall the dynamics of the system under con-

sideration

˙x = Ax + Bu + f (x, u).

First assume that x

∈ X

⊂ X , where X

is the set

of points in X such that the solution of the (discrete

time) optimization problem is given by

MPC

= −(B

A)x

,∀k ∈ N .

Implementation of Centralized MPC on the Quadruple-tank Process with Guaranteeing Stability

We note Q

the quadratic Lyapunov function associ-

ated with this K := (B

A). We deﬁne:

= {x ∈ X

x < c}. (7)

This set always exist and is not empty.

Lemma 1. there exists a h > 0 such that solving the

optimization problem (4) with any discretization time

h < h and without constraints leads to a controller K ,

and the solution to the associated solution to the Ric-

cati equation Q

such that (A − BK)

+ Q

(A −

BK) < −εI for some ε > 0.

Proof. This follows from (Kailath, 1980) Ch2 sec.

2.6. (see appendix).

Lemma 2. Considering the dynamical system (1)

with Lipschitz constant L on the set X, and e(t) =

x(t) − x

for t ∈ [0,h[ one has:

|e(t)| ≤

1 − Lh

|x(t)|.

Proof. The proof follows the same line as the one de-

veloped in (Tabuada, 2007) Event-triggered real-time

scheduling of stabilizing control tasks Theorem III.1.

(see appendix).

We deﬁne the set S

4|Q

= {x ∈ X

|γ(|x|) ≤

4|Q

}. We deﬁne the set

∗

= max

⊂ (X

∩ S

4|Q

). (8)

Such a set exist and has non empty interior. Note that

since X

and S

4|Q

contains an open neighborhood of

the origin so does P

∗

Theorem 1. For h <

L(8|Q

BK| + ε)

, x

∈ P

∗

sys-

tem (1) is locally exponentially stable under the model

predictive control policy deﬁned in (6).

Proof. We consider the e(t) = x(t) − x

for t ∈ [0,h[

Considering a Lyapunov function V (x) = x

∈ P

∗

We have

V (x) = ˙x

x + x

˙x,

V (x) =



Ax + Bu + f (x,u)



x + x



Ax + Bu + f (x,u)



Considering ﬁrst t ∈ [0,h[, given the fact that u =

−Kx

= −K(x − e) (with K = (B

A) )

V (x) =x

+ Q

+ e

x + x

+ f (x, K(e − x))

x + x

f (x,K(e − x)),

with A

= A − BK,D = BK.

From Lemma 1

V (x) ≤ −ε|x|

+ 2||Q

D|||x||e| + 2||Q

|||x|| f (x,K(e − x))|,

from A

and the deﬁnition of P

∗

| f (x,K(e − x))| ≤

4|Q

|x|,

from Lemma 2 and the deﬁnition of h

|e| ≤

8|Q

|x|.

Therefore for t ∈ [0,h[

V (x) ≤ −

|x|

So it follows by deﬁnition of P

∗

that at all time t ∈

[0,h[ x(t) ∈ P

∗

. Since x is continuous V (x) is also

continuous and x(h

) ∈ P

∗

. for all t ∈ [h,2 ∗ h[ The

control gain is given by u(t) = −Kx(h) and we can

use the previous argumentation iteratively. Deﬁning

lambda

max

) (resp. lambda

min

)) the biggest

(resp. smallest) eigenvalue of lambda

max

) We

have V (x) ≤ λ

max

)|x|

. One has by integrating

V (x(t)) ≤ −

2λ

max

)

V (x(t))

|x(t)| ≤ e

−

εt

4λ

max

)

max

)

min

)

One conclude that the original system is locally (i.e.

when x

∈ P

∗

) exponentially stable.

4 DESCRIPTION OF THE

BENCHMARK

A schematic diagram of the benchmark is shown in

Figure 1 (Alvarado et al., 2011). The objective of the

process is to control the water levels (h

) in the

lower tanks using two pumps.

The inputs of this process are pumps’ ﬂows

) and the outputs come from measuring the wa-

ter level in tanks (h

This model is identiﬁed by the following differential

equations

= −

2gh

= −

2gh

= −

2gh

1 − γ

= −

2gh

1 − γ

(9)

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

Figure 1: Johansson’s quadruple-tank process diagram.

where S(m

) is the cross-section of all four-tanks,

(m) and a

), i ∈ {1,2,3, 4} mention the water

level and the discharge constant of tank i, respectively.

A voltage is applied to pump j which provides the dis-

charge q

−1

) with the corresponding ratio of γ

and j ∈ {a, b}. g(ms

−2

) is denoted as the gravitational

acceleration. The parameter values are estimated ex-

perimentally in the laboratory and are presented in

Table1 (Alvarado et al., 2011).

Upon the four-tank process which has been il-

lustrated above as a benchmark, the centralized con-

troller (MPC) is tested, in order to analyze its proﬁ-

ciency. The model and controller is implemented in

SIMULINK and the MPC controller is computed us-

ing CVX (Boyd and Vandenberghe, 2004).

4.1 Prediction Model and Simulation

For applying the centralized MPC, there should be

a linear prediction model which is obtained through

linearizing (9) around an equilibrium point (h

The operating point is assigned from the equilibrium

levels presented in the previously referenced table.

Consider the variables around the operating points as

= h

− h

, i ∈ 1,2, 3, 4

= q

− q

= q

− q

Table 1: Parameters of the quadruple-tank.

Parameters Value Unit Description

1max

1.36 m Maximum level of the tank 1

2max

1.36 m Maximum level of the tank 2

3max

1.30 m Maximum level of the tank 3

4max

1.30 m Maximum level of the tank 4

min

0.2 m Minimum level in all cases

amax

3.26 m

/h Maximum ﬂow of q

bmax

4 m

/h Maximum ﬂow of q

min

0 m

/h Minimum ﬂow of q

and q

1.31e − 4 m

discharge constant of tank 1

1.51e − 4 m

discharge constant of tank 2

9.27e − 5 m

discharge constant of tank 3

8.82e − 5 m

discharge constant of tank 4

S 0.06 m

Cross-section of the tanks

0.3 Parameter of the 3-way valve

0.4 Parameter of the 3-way valve

0.65 m Linearization level of tank 1

0.66 m Linearization level of tank 2

0.65 m Linearization level of tank 3

0.66 m Linearization level of tank 4

1.63 m

/h Linearization ﬂow of q

2.00 m

/h Linearization ﬂow of q

The linearized continuous-time state-space model

becomes as

= A

x + B

u, (10)

in which, x = [x

] , u = [u

] , Q = C

R = I and the matrices are as follow







−1

τ1

τ3

−1

τ2

τ4

0 0

−1

τ3

0 0 0

−1

τ4













1−γ









1 0 0 0

0 1 0 0



with τ =

≥ 0 and i ∈ {1, 2,3,4}. For the im-

plementation of the centralized MPC, equation (10)

is discretized with a sampling time of Ts=5 seconds.

We have chosen the prediction horizon as Hp=20 sec-

onds.

4.2 Control Objectives

There are some issues which should be taken into con-

sideration before using the controller.

• Modelling: the class of a model is highly depen-

dant on the type of the controller which is going to

be used and also the aim of the controlling item.

For example, it is an important decision to use a

linear or nonlinear model.

• Targets: there are a variety of attributes assigned

by the employed speciﬁc type of the controller.

These attributes come as optimality, stability, fea-

sibility, etc.

Implementation of Centralized MPC on the Quadruple-tank Process with Guaranteeing Stability

• Required joint software: optimization routines,

simulation routines, etc.

5 RESULTS

The main purpose of the benchmark is keeping the

water levels of tanks 1 and 2 as close as possible

to their referenced levels. Thus, different reference

shifts are inserted in the benchmark used in (Alvarado

et al., 2011) to examine different equilibrium points.

As shown in Figures 2 and 3, the reference signals

change every 1000 seconds and the initial values are

according to the previously quoted operating points

in the linearization of (10).

Figure 2(a) displays the water level and the steady

state in tank 1. At the time 2000s, there is a differ-

ence of around 0.05m in the mentioned levels. Also,

at the time 3000s there is a similar deviation of 0.05m.

Figure 2(b) displays the water level and the steady

state in tank 2. In comparison to tank 1, The con-

vergence seems faster however some steady state er-

ror is present (unlike tank 1 where there is no steady

state error). At the time 2000s, there is a difference of

around 0.03m in the mentioned levels and at the time

3000s, this amount decreases up to 0.02m.

Figure 2: Water levels in tanks 1 (ﬁg. a)) and 2 (ﬁg. b)).

Figure 3, related to the tanks 3 and 4, reveals some

overshoots in water levels. For instance at around the

time of 2000s, tank 3 shows a sharp rise of the wa-

ter level reaches to 0.90m while the steady state at

this time is 0.85m. This tank displays a deep fall at

around the time 3000s and reaches the water level of

0.44m when the steady state at this time is 0.50m.

At around the time of 2000s, tank 4 shows a sharp rise

of the water level reaches to 0.91m while the steady

state at this time is 0.85m. This tank displays a deep

Figure 3: Water levels in tanks 3 (ﬁg. a)) and 4 (ﬁg. a)).

fall at around the time 3000s and reaches the wa-

ter level of 0.46m when the steady state at this time

is 0.51m. These overshoots are probably occurring

due to the points’ far distance from the linearization

points.

Moreover, the controller’s values are depicted in

Figure 3. Fig. 3(a) represents the ﬁrst control

in which the maximum allowable ﬂow is q

amax

3.26m

/h. Fig. 3(b) shows the second control in

which the maximum allowable ﬂow is q

bmax

= 4m

/h.

In overall, the saturation appears to be well handled

by the proposed MPC.

Therefore, the controller is generally able to deal with

the operational constraints. Although, the nonlinear

system was supposed to be locally exponentially sta-

ble, there were some inaccuracies possibly due to the

numerical errors or deﬁning the sampling time step in

the simulation step.

Figure 4: The controller’s values q

in ﬁg. a) and q

in ﬁg.

b).

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

6 CONCLUSION

In this work, the problem of MPC control design for a

four-tank benchmark model was considered. First, it

was shown that under suitable constraints the nonlin-

ear continuous time system can be stabilized by a lin-

ear discrete time controller. Moreover, we have con-

ducted simulations that show the good performance

of the control algorithm.

Further work will provide tighter estimates of the

region where stability and feasibility can be guaran-

teed. Another promising research direction is the use

of hybrid/multi model in order to enhance the control

performance and robustness.

ACKNOWLEDGEMENTS

The authors would like to thank the Regional Council

Hautes-de-France for its support.

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APPENDIX

Proof of Lemma 1

Proof. Solving the optimization problem (4) with

any discretization time h < h leads to an LQR con-

troller K associated with a Cost x

Such that

(A − BK)

(A − BK)− Q

= −Q.

Since A = I + hA + o(h),B = hB + o(h), deﬁning

W (h) := I + h(A − BK)

(W (h) + o(h))

(W (h) + o(h)) − Q

< −Q/2

h(A − BK +o(h))

+ hQ

(A − BK) + o(h) < −Q/2,

Therefore there exist h small enough such that

(A − BK)

+ Q

(A − BK) < −S,

Implementation of Centralized MPC on the Quadruple-tank Process with Guaranteeing Stability

With S a symmetric positive deﬁnite matrix therefore

(A − BK)

+ Q

(A − BK) < −λ

min

(S)I,

Proof of Lemma 2

Proof. For the dynamical system (1) with Lipschitz

constant L and considering the dynamics of

|e|

|x|

one has

|e|

|x|

1/2

|e|

|x|

−1/2

˙e(x

1/2

− (x

−1/2

˙x(e

1/2

|e|

|x|

˙x

|e||x|

−

˙x

|x||x|

|e|

|x|

|e|

|x|

≤

|e|| ˙x|

|e||x|

−

|x|| ˙x|

|x||x|

|e|

|x|

|e|

|x|

≤



1 +

|e|

|x|



| ˙x|

|x|

|e|

|x|

≤



1 +

|e|

|x|



L(|x| + |e|)

|x|

|e|

|x|

≤ L



1 +

|e|

|x|



integrating from time 0 to h the equation

φ = L(1 +

) with φ(0) = 0 one obtains

|e(t)| ≤

1 − Lh

|x(t)|.

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics