model reference control of lpv systems

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Journal of the Franklin Institute 346 (2009) 854–871

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Model reference control of LPV systems

Ali Abdullah�, Mohamed Zribi

Kuwait University, Electrical Engineering Department, P.O. Box 5969, Safat 13060, Kuwait

Received 2 June 2008; received in revised form 1 December 2008; accepted 26 April 2009

Abstract

This paper deals with the problem of model reference control for linear parameter varying (LPV)

systems. The LPV systems under consideration depend on a set of parameters that are bounded and

available online. The main contribution of this paper is to design an LPV model reference control

scheme for LPV systems whose state-space matrices depend affinely on a set of time-varying

parameters that are bounded and available online. The design problem is divided into two

subproblems: the design of the coefficient matrices of the controller and the design of the gain of the

state feedback controller for LPV systems. The singular value decomposition is used to obtain the

coefficient matrices, while the linear matrix inequality methodology is used to obtain the parameter-

dependent state feedback gain of the control scheme. A simple numerical example is used to illustrate

the proposed design and a coupled-tank process example is used to demonstrate the usefulness and

practicality of the proposed design. Simulation and experimental results indicate that the proposed

scheme works well.

r 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Keywords: Model reference control; Linear parameter varying system; Linear matrix inequality; Singular value

decomposition; Coupled-tank process

1. Introduction

Linear parameter varying systems are a class of linear systems whose state-spacematrices depend on a set of time-varying parameters that are not known in advance, but itcan be measured or estimated upon operation of the system. The idea of controlling LPVsystems has been introduced in [1–4], then further extended during the last two decades to

2.00 r 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

.jfranklin.2009.04.006

nding author. Tel.: +965 24987364; fax: +965 24817451.

dress: [email protected] (A. Abdullah).

www.elsevier.com/locate/jfranklin

dx.doi.org/10.1016/j.jfranklin.2009.04.006

mailto:[email protected]

ARTICLE IN PRESSA. Abdullah, M. Zribi / Journal of the Franklin Institute 346 (2009) 854–871 855

produce many control methods. Examples of these methods include: observer [5,6], statefeedback controllers [7], state feedback controllers with LQG performance [8], statefeedback controllers with H2 performance [9,10], state feedback controllers with H1performance [11], output feedback controllers with LQG performance [12], H1 controllers[13–21], output feedback controllers subject to control saturation [22], anti-windupcontrollers [23,24], model predictive controllers [25,26], adaptive neural networkcontrollers [27], and fuzzy controllers [28,29].

Most of the abovementioned techniques have been applied to practical systems. Controldesigns for LPV systems such that missiles [30–35], aircrafts and spacecrafts [36–39],energy production systems [40–46], inverted pendulums [47], automated vehicles [48],winding systems [49], wafer scanners [50], robotic systems [51], and congestion incomputer-networks and web servers [52,53] have been investigated.

The objective of this paper is to design an LPV model reference control scheme for LPVsystems whose state-space matrices depend affinely on a set of time-varying parameters that arebounded and available online. The proposed LPV model reference control is an extension to thewell-known linear time-invariant (LTI) model reference controller thathas been extensively studied by many researchers, see for example [54–57] and the referencestherein.

This paper is organized as follows. The second section formulates the problem underinvestigation. In the following section, the structure of the model reference control ispresented using a set of matrix equalities and inequalities which need to be solved for theparameters of the control scheme. The fourth section presents a methodology for thedesign of the control parameters. In Section 5, the proposed model reference control isapplied to a numerical example and to a coupled-tank process; the simulation andexperimental results are presented and discussed. Finally, some concluding remarks aregiven in Section 6.

2. Problem formulation

This paper considers the model reference control of a class of linear parameter varying(LPV) systems described by the following equations:

_xðtÞ ¼ Að/ðtÞÞxðtÞ þ BuðtÞ, (1)

yðtÞ ¼ CxðtÞ, (2)

where

Að/ðtÞÞ ¼ A0 þXN

i¼1

fiðtÞAi.

The matrices Ai (i ¼ 0; 1; . . . ;N), B and C are known constant matrices of appropriatedimensions. The vector xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the input vector andyðtÞ 2 Rp is the output vector. The vector /ðtÞ ¼ ½f1ðtÞ f2ðtÞ . . . fNðtÞ�

T 2 RN is the time-varying parameter vector with N being the number of time-varying parameters.

The following assumptions regarding the LPV system (1) and (2) are made:

Assumption 1. The state vector xðtÞ is measurable or it can be estimated online.

Assumption 2. The parameter vector /ðtÞ is measurable or it can be estimated online.

ARTICLE IN PRESSA. Abdullah, M. Zribi / Journal of the Franklin Institute 346 (2009) 854–871856

Assumption 3. The ith element, fiðtÞ, of the parameter vector /ðtÞ is assumed to varybetween known f

iand fi, i.e., fi

� fiðtÞ � fi.

Assumption 4. The matrix C is a full row rank matrix, i.e., the inverse of CCT exists.

Remark 1.

�
If the state vector xðtÞ cannot be measured, the estimate of xðtÞ can be obtained onlineusing the results in [5]. � Assumption 2 is essential in the LPV synthesis. Indeed, many practical systems can be
modeled as LPV systems whose parameter vector can be measured or estimated, see forexample [30–53].
� The matrices B and C in (1)–(2) are assumed to be constant. When this is not the case,
low-pass filters with sufficiently large bandwidth can be used to filter the system’s inputsand outputs and hence to move all the time-varying parameters into the state matrix, see[16]. Therefore, even in the case when B and C are functions of fiðtÞ, the augmentedmodel of the system can be converted into the form (1)–(2).
� Assumption 4 means that the sensor measurements are independent.
The objective of the paper is to design a model reference controller for the LPV system(1)–(2) such that the system output yðtÞ converges to the desired reference output yðtÞasymptotically, where the reference output yðtÞ is assumed to be obtained from thefollowing LPV reference model:

_xðtÞ ¼ Að/ðtÞÞxðtÞ þ BuðtÞ, (3)

yðtÞ ¼ CxðtÞ, (4)

where

Að/ðtÞÞ ¼ A0 þXN

i¼1

fiðtÞAi.

The matrices Ai (i ¼ 0; 1; . . . ;N), B and C are known constant matrices of appropriatedimensions. The vector xðtÞ 2 Rq is the reference state vector, uðtÞ 2 R‘ is the referenceinput vector and yðtÞ 2 Rp is the output vector of the reference model.

Remark 2. When the matrices Ai ¼ 0 for i ¼ 1; 2; . . . ;N, the LPV reference model (3)–(4)becomes an LTI reference model. Therefore, the LTI reference model is a special class ofthe LPV reference model.

Note that in the rest of the paper, the variable dependence on time will be suppressedwhen no confusion might arise. Also, the symbol % will be used to represent the transposeof a matrix G in the following symmetric matrix:

Gþ %:¼Gþ GT .

3. Model reference control for LPV systems

The structure of the model reference control scheme is chosen to be similar to the well-known structure for the LTI case [56] except that some of the design gain matrices areparameter-dependent.


Consider the control law given by

u ¼ Kð/Þ½x�Gx� þMð/ÞxþQu, (5)

where the matrices G, Q, Mð/Þ and Kð/Þ are design matrices of appropriate dimensionssatisfying the following relations:

C ¼ CG, (6)

GB ¼ BQ, (7)

GAð/Þ ¼ BMð/Þ þ Að/ÞG, (8)

Kð/Þ ¼ Lð/ÞP�1, (9)

with Að/Þ, B, C, Að/Þ, B, and C are specified in (1)–(2) and (3)–(4). The matrix Lð/Þ andthe positive definite symmetric matrix P in Eq. (9) are the solutions of the following matrixinequality:

Hð/Þ:¼Að/ÞPþ BLð/Þ þ %o0. (10)

Remark 3. It should be pointed out that the structure of Lð/Þ has to be chosen in order tosynthesize the control law. Since Að/Þ in Eq. (1) depends affinely on the parameter vector/, the structure of Lð/Þ is chosen to have the same structure of Að/Þ, that is

Lð/Þ ¼ L0 þXN

i¼1

fiLi,

where Li’s are design constant matrices.

The following lemma shows that by using the control law (5), the system output yconverges to the reference output y asymptotically.

Lemma 1. The control law (5) applied to the LPV system (1)–(2) guarantees that the system

output yðtÞ converges to the reference output yðtÞ asymptotically.

Proof. Define the state error e ¼ x�Gx and the output error ey ¼ y� y.Using (1), (3), (5), (7) and (8), it follows that:

_e ¼ Að/Þx�GAð/Þxþ Bu�GBu

¼ Að/Þxþ BKð/Þ½x�Gx� þ ½BMð/Þ �GAð/Þ�xþ ½BQ�GB�u

¼ ½Að/Þ þ BKð/Þ�e. (11)

Furthermore, using (2), (4), and (6), yields the following:

ey ¼ y� y

¼ Cx� Cx

¼ Cx� CGx

¼ Ce.

Consider the following Lyapunov function candidate:

W ¼ eTP�1e. (12)


Taking the derivative of Eq. (12) with respect to time and using Eq. (11), the condition_Wo0 can be written as

ATð/ÞP�1 þ P�1Að/Þ þ KT ð/ÞBTP�1 þ P�1BKð/Þo0. (13)

Then pre-multiplying and post-multiplying both sides of inequality (13) by P and usingLð/Þ ¼ Kð/ÞP, one obtains Hð/Þo0.Hence, it can be concluded that the state error e and the output error ey ¼ Ce converge

to zero asymptotically. &

In order to implement the model reference controller given by Eq. (5), one needs thecoefficient matrices G, Q and Mð/Þ as well as the controller gain matrix Kð/Þ. The designof these parameters will be discussed in the next section.

4. Design of the parameters of the model reference controller

4.1. Parametrization of the controller’s coefficient matrices G, Q and MðfÞ

In this subsection, a parametrization of the coefficient matrices G, Q and Mð/Þ neededfor the design of the controller (5) is presented.Let I be the identity matrix of appropriate dimensions.

Theorem 1. Given that the singular value decomposition of the input matrix B is such that

B ¼ USWT , (14)

with UTU ¼ UUT ¼ I, WTW ¼WWT ¼ I, and S 2 Rn�m with

UT ¼U1m�n

U2ðn�mÞ�n

" #; S ¼

S

0ðn�mÞ�m

" #; S ¼ diag½s1 s2 . . . sm�,

where s1 � s2 � � � � � sm40 are the singular values of the input matrix B.Then, there exist matrices G, Q and Mð/Þ satisfying the matrix equations (6)–(8) if and

only if there exists a matrix F 2 Rn�q satisfying the following relations:

U2GB ¼ 0, (15)

U2½GAð/Þ � Að/ÞG� ¼ 0, (16)

where

G ¼ CT½CCT

��1Cþ ½I� CT½CCT

��1C�F. (17)

Furthermore, the matrices Q and Mð/Þ are given by

Q ¼WS�1U1GB, (18)

Mð/Þ ¼WS�1U1½GAð/Þ � Að/ÞG�. (19)

Proof.

�
Notice that the general solution of G in Eq. (6) is given by Eq. (17) with F being anarbitrary matrix such that
C ¼ CG¼)

G ¼ CT½CCT


��1C�F.


Hence, one can easily obtain Eq. (17).
� Using the singular value decomposition of B into Eq. (7) and then pre-multiplying both
sides by UT , we get

GB ¼ BQ()

GB ¼ USWTQ¼)

UTGB ¼ UTUSWTQ()

UTGB ¼ SWTQ.

Now using

UT ¼U1

U2

" #and S ¼

S

0

� �,

yields the following:

U1

U2

" #GB ¼

S

0

� �WTQ.

which implies that

U1GB ¼ SWTQ and U2GB ¼ 0.

Note that U1GB ¼ SWTQ implies that Q ¼WS�1U1GB.Hence Eqs. (15) and (18) are proved.
� Using the singular value decomposition of B into Eq. (8) and then pre-multiplying both
sides by UT , we get

GAð/Þ ¼ BMð/Þ þ Að/ÞG()

GAð/Þ ¼ USWT Mð/Þ þ Að/ÞG¼)

UTGAð/Þ ¼ UTUSWT Mð/Þ þUT Að/ÞG()

UTGAð/Þ ¼ SWT Mð/Þ þUT Að/ÞG.

Now using

UT ¼U1

U2

" #and S ¼

S

0

� �,

yields the following:

U1

U2

" #GAð/Þ ¼

S

0

� �WT Mð/Þ þ

U1

U2

" #Að/ÞG.

which implies that

U1GAð/Þ ¼ SWT Mð/Þ þU1Að/ÞG (20)

and

U2GAð/Þ ¼ U2Að/ÞG. (21)


The first result given by Eq. (20) implies that

Mð/Þ ¼WS�1U1½GAð/Þ � Að/ÞG�.

The second result given by Eq. (21) implies that

U2½GAð/Þ � Að/ÞG� ¼ 0.

Hence, Eqs. (16) and (19) are proved.Therefore, Theorem 1 which gives a parametrization of the coefficient matrices G, Q andMð/Þ is proved. &4.2. Computation of the gain matrix KðfÞ In order to compute thegain matrix Kð/Þ ¼ Lð/ÞP�1, the solution (Lð/Þ and P) of the matrix inequality problem(10) is presented next. Lemma 2. Consider the parameter vector / that is varying inside a

hyper-rectangle with 2N vertices defined as

Vj 2 fðv1;j ; . . . ; vN;jÞjvi;j 2 ffi;figg; j ¼ 1; . . . ; 2N ,

where vi;j 2 R is the ith element of Vj 2 RN .

The following statements are equivalent:
� Hð/Þo0 for all possible values of /, � HðVjÞo0 for j ¼ 1; . . . ; 2N . Proof. See [16]. &
Using Lemma 2, the matrix inequality problem (10) can be formulated as a finite matrixinequality problem which can be solved efficiently using an LMI numerical algorithm [58].Once Li (i ¼ 0; 1; . . . ;N), and P satisfying HðVjÞo0 for j ¼ 1; . . . ; 2N are found, the gain

Kð/Þ can be computed using Eq. (9) as follows:

Kð/Þ ¼ L0 þXN

i¼1

fiLi

" #P�1. (22)

4.3. Procedure for designing the model reference controller

To summarize the results of the last two subsections, the following steps can beperformed to compute the coefficients matrices G, Q and Mð/Þ and the gain Kð/Þ of thecontrol law (5) for the LPV system (1)–(2) and the LPV reference model (3)–(4).

Step 1: Write the matrix G given in Eq. (17) as a function of F.Step 2: Find the matrices U1, U2, S, and W by carrying out the singular value

decomposition (14) for the input matrix B.Step 3: Substitute the matrices G and U2 into the matrix equations (15) and (16) to getðn�mÞ � ‘ equations from Eq. (15) and ðn�mÞ � q equations from Eq. (16). Then, solvethese equations for the n� q elements of the matrix F. If the matrix F does not exist, thenthe proposed model reference control cannot be designed.

Step 4: Compute the coefficients matrices G, Q and Mð/Þ by substituting the matrix F

into Eq. (17), and then substituting the result and the matrices U1, S, and W into the Eqs.(18) and (19).


Step 5: Find the vertices Vj ðj ¼ 1; . . . ; 2N Þ, then solve 2N matrix inequalities HðVjÞo0

for the matrices Li (i ¼ 0; 1; . . . ;N), and P. After that, use Eq. (22) to compute the gainKð/Þ.

5. Numerical examples

In this section, the design procedure is illustrated on the following two examples.

5.1. First example

Consider the LPV system (1)–(2) with state-space matrices

Að/Þ ¼f1 �1

1 0

� �; B ¼

1

0

� �; C ¼ ½0 1�,

where �1 ¼ f1� f1:¼ cosðyÞ � f1 ¼ 1.

Also, consider the reference model (3)–(4) with state-space matrices

Að/Þ ¼�2:8 �2

1 0

� �; B ¼

2

0

� �; C ¼ ½0 1�.

The following steps are performed to obtain the control law parameters.Step 1: Let

F ¼f 1 f 2

f 3 f 4

" #,

then the matrix G in Eq. (6) is obtained using Eq. (17) such that

G ¼ CT½CCT


��1C�F ¼f 1 f 2

0 1

� �,

where f 1 and f 2 will be specified later on, and f 3 and f 4 are chosen to be zero.Step 2: The singular value decomposition of the input matrix B is such that B ¼ USWT

where

UT ¼1 0

0 1

� �¼

U1

U2

" #; S ¼

1

0

� �¼

S

0

� �; W ¼ 1,

with

U1 ¼ ½1 0�; U2 ¼ ½0 1�; S ¼ 1.

Step 3: Using the matrices G and U2, it can be shown that Eqs. (15) and (16) are satisfiedwhen the elements of the matrix F are: f 1 ¼ 1 and f 2 ¼ f 3 ¼ f 4 ¼ 0.

Step 4: Using the results in Steps 1–3, the coefficient matrices G, Q and Mð/Þ arecalculated from (17)–(19) as follows:

G ¼1 0

0 1

� �; Q ¼ 2; Mð/Þ ¼ ½�2:8� f1 � 1�.

ARTICLE IN PRESS

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4

Time, s

Ref

eren

ce m

odel

and

sys

tem

out

puts

Output of the reference modelOutput of the system

Fig. 1. Simulation results for the first model reference control system.

A. Abdullah, M. Zribi / Journal of the Franklin Institute 346 (2009) 854–871862

Step 5: The vertices V1 ¼ f1 ¼ 1 and V2 ¼ f1¼ �1 are used to solve the following matrix

inequalities:

HðV1Þ ¼ Aðf1ÞPþ B½L0 þ f1L1� þ %o0,

HðV2Þ ¼ Aðf1ÞPþ B½L0 þ f

1L1� þ %o0.

The following results are obtained using the LMI Control Toolbox [59]:

L0 ¼ ½�73:7192 0�; L1 ¼ ½�81:9103 32:7641�,

P ¼81:9103 �32:7641

�32:7641 81:9103

� �.

Then, Eq. (22) is used to compute the following controller gain:

Kð/Þ ¼ ½�1:0714� f1 � 0:4286�.

The designed model reference controller is applied to the LPV system. Fig. 1 shows thesimulation results when the initial conditions are taken to be zero and the reference input ischosen as uðtÞ ¼ 1. It can be seen from Fig. 1 that the system output, yðtÞ, is almostidentical to the reference output, yðtÞ. We expect to get this result, because the originaldynamic model is used to design the controller.

5.2. Second example

In this subsection, the model reference control scheme developed in the previous sectionsis applied to a coupled-tank process where the process is modeled as an LPV system. Thecoupled-tank process used in this article is designed by Quanser Inc. [60].The coupled-tank process is composed of two cylindrical tanks: an upper tank (tank 1)

and a lower tank (tank 2), see Fig. 2. A pump is used to thrust water from the waterreservoir to tank 1 and the outflow of tank 1 flows through tank 2 to the water reservoir.

ARTICLE IN PRESS

pk u

Water Reservoir

Inlet Pipe

Tank 1

1h

Tank 2

2hPump

Fig. 2. Schematic diagram of the coupled-tank process.

A. Abdullah, M. Zribi / Journal of the Franklin Institute 346 (2009) 854–871 863

Pressure sensors located at the bottom of each tank are used to measure the water levels inthe tanks. The objective is to design a model reference controller so that the water level intank 2 tracks the output of a reference model.

The dynamics model of the water levels h1ðtÞ and h2ðtÞ can be written as [61]

_h1ðtÞ ¼ �a1

A1

ffiffiffiffiffi2g

p ffiffiffiffiffiffiffiffiffiffih1ðtÞ

pþ

kp

A1uðtÞ, (23)

_h2ðtÞ ¼a1

A2

ffiffiffiffiffi2g


p�

a2

A2

ffiffiffiffiffi2g


p, (24)

yðtÞ ¼ h2ðtÞ, (25)

where hi is the water level in tank i; Ai the cross-section area of tank i; ai the cross-sectionarea of tank i outflow orifice; u the voltage applied to pump; kp the gain of pump; g thegravitational constant. The physical quantities are given as follows:

A1 ¼ A2 ¼ 15:5179 cm2; a1 ¼ a2 ¼ 0:1781 cm2,

kp ¼ 3:3 cm3=Vs; g ¼ 981 cm=s2.

In order to apply the proposed model reference control scheme to the coupled-tank process(23)–(25), an LPV model of the coupled-tank process is derived as follows.

First, a standard polynomial fitting technique [62] is used to approximateffiffiffiffihi

pfor

0 � hi � 30 cm with fihi, where

fi ¼ a4h4i þ a3h

3i þ a2h

2i þ a1hi þ a0,


with

a4 ¼ 2:981� 10�7; a3 ¼ �3:659� 10�5; a2 ¼ 1:73� 10�3,

a1 ¼ �4:036� 10�2; a0 ¼ 0:583.

It can be shown that the parameters f1 and f2 are bounded such that 0:1 ¼ f1� f1 �

f1 ¼ 0:6 and 0:1 ¼ f2� f2 � f2 ¼ 0:6.

Then, the dynamic equations of the water levels are written in the LPV form (1)–(2) as

_x ¼ Að/Þxþ Bu, (26)

y ¼ Cx, (27)

where

x:¼h1

h2

" #; Að/Þ ¼

�0:5085f1 0

0:5085f1 �0:5085f2

" #; B ¼

0:2127

0

� �; C ¼ ½0 1�.

Remark 4. An LTI reference model (i.e., the reference model (3)–(4) with Ai ¼ 0 fori ¼ 1; 2; . . . ;N) cannot be designed for the coupled-tank system because it can be shownthat Eq. (16) reduces to

C A0 � 0:5085f1F1 þ 0:5085f2C ¼ 0, (28)

where F1 is the first row of the matrix F. Eq. (28) is satisfied for all possible values of f1 and f2

when C ¼ F1 ¼ 0 which is unacceptable solution. Therefore, in this case an LPV referencemodel gives the designer more freedom to design the proposed model reference controller.

The structure of the LPV reference model is depicted in Fig. 3 where an integrator hasbeen introduced to ensure zero steady-state error. The state-space equations of the LPVreference model can be written in the form (3)–(4) as

_x ¼ Að/Þxþ Bu,

y ¼ Cx,

with

x ¼x1

x2

" #; Að/Þ ¼

a1ð/Þ a2ð/Þ

�bc 0

� �; B ¼

0

b

� �; C ¼ ½c 0�,

where a1ð/Þ, a2ð/Þ, b, and c will be specified later on.The following steps are performed to obtain the control law parameters.Step 1: Let

F ¼f 1 f 2

f 3 f 4

" #,

c1 1 1 2

1

x =a ( ) x +a ( ) x

y= x2u b ( ) d t 2x

y

Fig. 3. Structure of the reference model.


then the matrix G in Eq. (6) is obtained using Eq. (17) such that

G ¼ CT½CCT


��1C�F ¼f 1 f 2

c 0

� �,

where f 1 and f 2 will be specified later on, and f 3 and f 4 are chosen to be zero.Step 2: The singular value decomposition of the input matrix B is such that B ¼ USWT

where

UT ¼1 0

0 1

� �¼

U1

U2

" #; S ¼

0:2127

0

� �¼

S

0

� �; W ¼ 1,

with

U1 ¼ ½1 0�; U2 ¼ ½0 1�; S ¼ 0:2127.

Step 3: Using the matrices G and U2, it can be shown that Eq. (15) is satisfied for any F

and Eq. (16) is given by

½cða1ð/Þ þ 0:5085f2Þ � 0:5085f 1f1 ca2ð/Þ � 0:5085f 2f1� ¼ 0,

which implies that

a1ð/Þ ¼0:5085f 1

cf1 � 0:5085f2,

a2ð/Þ ¼0:5085f 2

cf1.

The gains f 1 and f 2 need to be properly designed in order to guarantee the stability of theLPV reference model of the coupled-tank process for all admissible values of f1 and f2.One way to solve the above problem is to assume the values of f 1 and f 2 and then solve thefollowing set of LMIs for a positive definite symmetric matrix P [17]:

ATðVjÞPþ P AðVjÞo0 for j ¼ 1; 2; 3; 4.

The gains f 1 ¼ 0:01 and f 2 ¼ 0:05 are chosen and the stability is checked using the LMIControl Toolbox [59] when b ¼ c ¼ 1.

Remark 5. If the transient response of the reference model needs to be improved, anadditional set of LMIs can be formulated to ensure that the poles of the reference model lieinside a selected sub-region of the left-half plane for all admissible values of fi; furtherdetails can be found in Ref. [18].

Step 4: Using the results in Steps 1–3, the coefficient matrices G, Q and Mð/Þ arecalculated from (17)–(19) as follows:

G ¼0:01 0:05

1 0

� �; Q ¼ 0:23507,

Mð/Þ ¼ ½�0:23507þ 0:024147f1 � 0:023907f2 0:12073f1�.

Step 5: The following vertices:

V1 ¼f1

f2

" #¼

0:6

0:6

� �; V2 ¼

f1

f2

" #¼

0:6

0:1

� �,


V3 ¼f1

f2

" #¼

0:1

0:6

� �; V4 ¼

f1

f2

" #¼

0:1

0:1

� �,

are used to solve the matrix inequalities:

HðV1Þ ¼ AðV1ÞPþ B½L0 þ f1L1 þ f2L2� þ %o0,



HðV4Þ ¼ AðV4ÞPþ B½L0 þ f1L1 þ f

2L2� þ %o0.

The following results are obtained using the LMI Control Toolbox [59]:

L0 ¼ ½�3:5255 0�; L1 ¼ ½3:5854 � 4:5817�,

L2 ¼ ½0 � 0:99626�; P ¼1:4997 �0:41673

�0:41673 1:9394

� �.

Then, Eq. (22) is used to compute the controller gain as

Kð/Þ ¼ ½�2:5þ 1:8444f1 � 0:1518f2 � 0:53717� 1:966f1 � 0:5463f2�.

The designed model reference controller is applied to the nonlinear model of the coupled-tank process. Fig. 4 shows the simulation results when the initial water level in tank 2 is0 cm. The objective is to force the water level in tank 2 to converge to the desired referenceoutput with a steady state value of 10 cm. It can be seen from Fig. 4 that the water level intank 2, h2, follows the desired output of the reference model, y, very closely. Also, theoutput error e ¼ h2 � y is shown in Fig. 5. The result shows that the steady state outputerror is within 0:2mm. This nonzero steady state output error might be attributed to thediscrepancies between the dynamics model of the process (23)–(25) and the LPV model(26) and (27) which is used to design the controller.

0 50 100 150 200 250 300 350 4000

2

4

6

8

10

12

Time, s

Wat

er L

evel

in T

ank

2, c

m

Water level in tank 2Output of the reference model

Fig. 4. Simulation results for the water level in tank 2.

ARTICLE IN PRESS

0 50 100 150 200 250 300 350 400−2

0

2

4

6

8

10

12

Time, s

Out

put E

rror

, cm

Fig. 5. Simulation results for the output error e ¼ h2 � y.

Fig. 6. The coupled-tank process.

A. Abdullah, M. Zribi / Journal of the Franklin Institute 346 (2009) 854–871 867

The designed model reference controller was next implemented on an actual coupled-tank process depicted in Fig. 6. The results are presented in Figs. 7 and 8. Fig. 7 shows thatthe water level in tank 2 follows very closely the desired output of the reference model.Note that the steady state output error is within 0:8mm (see Fig. 8). The experimental

ARTICLE IN PRESS

0 50 100 150 200 250 300 350 4000

2

4

6

8

10

12

Time, s

Wat

er L

evel

in T

ank

2, c

m

Water level in tank 2Output of the reference model

Fig. 7. Experimental results for the water level in tank 2.

0 50 100 150 200 250 300 350 400−2

0

2

4

6

8

10

12

Time, s

Out

put E

rror

, cm

Fig. 8. Experimental results for the output error e ¼ h2 � y.

A. Abdullah, M. Zribi / Journal of the Franklin Institute 346 (2009) 854–871868

steady state output error is larger than the steady state error obtained numerically due toseveral factors such as: the unmodeled dynamics of the pump, the time delay, the sensorbias, the sensor noise, and the uncertainty on the parameters of the process.Therefore, it can be concluded that the designed model reference control scheme works

well for the coupled-tank process.

6. Conclusion

In this work, an LPV model reference controller is designed for linear parameter varying(LPV) systems. The controller is carried out by: solving a set of matrix equations using the


singular value decomposition of the input matrix, and then obtaining a parameter-dependent state feedback gain using linear matrix inequalities. A simple numerical exampleis used to illustrate the proposed design. To demonstrate the usefulness and the practicalityof the proposed control scheme, the model reference control of a coupled-tank process isstudied in details. The coupled-tank process is first modeled as an LPV system, then thedetailed derivation of the controller is given. The controlled system is simulated and thenimplemented. The simulation as well as the experimental results indicate that the proposedcontrol scheme works well. Therefore, the proposed scheme can be used to design modelreference controllers for LPV systems.

Future work will attempt to design robust LPV model reference controllers.

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