- Email: [email protected]
Model reference control of LPV systems
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Journal of the Franklin Institute 346 (2009) 854–871 www.elsevier.com/locate/jfranklin
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 parameterdependent 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-space matrices depend on a set of time-varying parameters that are not known in advance, but it can be measured or estimated upon operation of the system. The idea of controlling LPV systems has been introduced in [1–4], then further extended during the last two decades to Corresponding author. Tel.: +965 24987364; fax: +965 24817451.
E-mail address: [email protected] (A. Abdullah). 0016-0032/$32.00 r 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2009.04.006
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produce many control methods. Examples of these methods include: observer [5,6], state feedback controllers [7], state feedback controllers with LQG performance [8], state feedback controllers with H2 performance [9,10], state feedback controllers with H1 performance [11], output feedback controllers with LQG performance [12], H1 controllers [13–21], output feedback controllers subject to control saturation [22], anti-windup controllers [23,24], model predictive controllers [25,26], adaptive neural network controllers [27], and fuzzy controllers [28,29]. Most of the abovementioned techniques have been applied to practical systems. Control designs 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 in computer-networks and web servers [52,53] have been investigated. The objective 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 proposed LPV model reference control is an extension to the well-known linear time-invariant (LTI) model reference controller that has been extensively studied by many researchers, see for example [54–57] and the references therein. This paper is organized as follows. The second section formulates the problem under investigation. In the following section, the structure of the model reference control is presented using a set of matrix equalities and inequalities which need to be solved for the parameters of the control scheme. The fourth section presents a methodology for the design of the control parameters. In Section 5, the proposed model reference control is applied to a numerical example and to a coupled-tank process; the simulation and experimental results are presented and discussed. Finally, some concluding remarks are given 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: _ ¼ Að/ðtÞÞxðtÞ þ BuðtÞ, xðtÞ
(1)
yðtÞ ¼ CxðtÞ,
(2)
where Að/ðtÞÞ ¼ A0 þ
N X
fi ðtÞAi .
i¼1
The matrices Ai (i ¼ 0; 1; . . . ; N), B and C are known constant matrices of appropriate dimensions. The vector xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the input vector and yðtÞ 2 Rp is the output vector. The vector /ðtÞ ¼ ½f1 ðtÞ f2 ðtÞ . . . fN ðtÞT 2 RN is the timevarying 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.
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Assumption 3. The ith element, fi ðtÞ, of the parameter vector /ðtÞ is assumed to vary between known fi and fi , i.e., fi fi ðtÞ 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 online using 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 for example [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 inputs and 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 fi ðtÞ, the augmented model 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 yðtÞ asymptotically, where the reference output yðtÞ is assumed to be obtained from the following LPV reference model: _ ¼ Að/ðtÞÞxðtÞ þ BuðtÞ, xðtÞ
(3)
yðtÞ ¼ CxðtÞ,
(4)
where Að/ðtÞÞ ¼ A0 þ
N X
fi ðtÞAi .
i¼1
The matrices Ai (i ¼ 0; 1; . . . ; N), B and C are known constant matrices of appropriate dimensions. The vector xðtÞ 2 Rq is the reference state vector, uðtÞ 2 R‘ is the reference input 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 of the LPV reference model. Note that in the rest of the paper, the variable dependence on time will be suppressed when no confusion might arise. Also, the symbol % will be used to represent the transpose of a matrix G in the following symmetric matrix: G þ %:¼G þ GT .
3. Model reference control for LPV systems The structure of the model reference control scheme is chosen to be similar to the wellknown structure for the LTI case [56] except that some of the design gain matrices are parameter-dependent.
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Consider the control law given by u ¼ Kð/Þ½x Gx þ Mð/Þx þ Qu,
(5)
where the matrices G, Q, Mð/Þ and Kð/Þ are design matrices of appropriate dimensions satisfying the following relations: C ¼ CG,
(6)
GB ¼ BQ,
(7)
GAð/Þ ¼ BMð/Þ þ Að/ÞG, 1
Kð/Þ ¼ Lð/ÞP ,
(8) (9)
with Að/Þ, B, C, Að/Þ, B, and C are specified in (1)–(2) and (3)–(4). The matrix Lð/Þ and the positive definite symmetric matrix P in Eq. (9) are the solutions of the following matrix inequality: Hð/Þ:¼Að/ÞP þ BLð/Þ þ %o0.
(10)
Remark 3. It should be pointed out that the structure of Lð/Þ has to be chosen in order to synthesize 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 þ
N X
fi Li ,
i¼1
where Li ’s are design constant matrices. The following lemma shows that by using the control law (5), the system output y converges 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/Þx þ Bu GBu ¼ Að/Þx þ BKð/Þ½x Gx þ ½BMð/Þ GAð/Þx þ ½BQ GBu ¼ ½Að/Þ þ BKð/Þe.
(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 eT P1 e.
(12)
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Taking the derivative of Eq. (12) with respect to time and using Eq. (11), the condition _ Wo0 can be written as AT ð/ÞP1 þ P1 Að/Þ þ K T ð/ÞBT P1 þ P1 BKð/Þo0.
(13)
Then pre-multiplying and post-multiplying both sides of inequality (13) by P and using Lð/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 the coefficient matrices G, Q and Mð/Þ as well as the controller gain matrix Kð/Þ. The design of 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 needed for 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 , T
(14)
T
T
T
nm
with U U ¼ UU ¼ I, W W ¼ WW ¼ I, and S 2 R with " # " # U1mn S UT ¼ ; S¼ ; S ¼ diag½s1 s2 . . . sm , U2ðnmÞn 0ðnmÞm where s1 s2 sm 40 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 Rnq satisfying the following relations: U2 GB ¼ 0,
(15)
U2 ½GAð/Þ Að/ÞG ¼ 0,
(16)
where G ¼ CT ½CCT 1 C þ ½I CT ½CCT 1 CF.
(17)
Furthermore, the matrices Q and Mð/Þ are given by Q ¼ WS1 U1 GB, 1
Mð/Þ ¼ WS U1 ½GAð/Þ Að/ÞG.
(18) (19)
Proof. Notice that the general solution of G in Eq. (6) is given by Eq. (17) with F being an arbitrary matrix such that C ¼ CG¼) G ¼ CT ½CCT 1 C þ ½I CT ½CCT 1 CF.
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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 USWT Q¼) UT GB ¼ UT USWT Q() UT GB ¼ SWT Q. Now using
"
T
U ¼
U1
# and
U2
S S¼ , 0
yields the following: " # U1 S WT Q. GB ¼ U2 0 which implies that U1 GB ¼ SWT Q
and
U2 GB ¼ 0.
Note that U1 GB ¼ SWT Q implies that Q ¼ WS1 U1 GB. 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) UT GAð/Þ ¼ UT USWT Mð/Þ þ UT Að/ÞG() UT GAð/Þ ¼ SWT Mð/Þ þ UT Að/ÞG. Now using T
U ¼
"
U1
#
U2
and
S¼
S , 0
yields the following: " # " # U U1 S 1 WT Mð/Þ þ GAð/Þ ¼ Að/ÞG. U2 U2 0 which implies that U1 GAð/Þ ¼ SWT Mð/Þ þ U1 Að/ÞG
(20)
U2 GAð/Þ ¼ U2 Að/ÞG.
(21)
and
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The first result given by Eq. (20) implies that Mð/Þ ¼ WS1 U1 ½GAð/Þ Að/ÞG. The second result given by Eq. (21) implies that U2 ½GAð/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 and Mð/Þ is proved. &4.2. Computation of the gain matrix KðfÞ In order to compute the gain matrix Kð/Þ ¼ Lð/ÞP1 , 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;j Þjvi;j 2 ffi ; fi gg;
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 /, HðVj Þo0 for j ¼ 1; . . . ; 2N . Proof. See [16]. &
Using Lemma 2, the matrix inequality problem (10) can be formulated as a finite matrix inequality problem which can be solved efficiently using an LMI numerical algorithm [58]. Once Li (i ¼ 0; 1; . . . ; N), and P satisfying HðVj Þo0 for j ¼ 1; . . . ; 2N are found, the gain Kð/Þ can be computed using Eq. (9) as follows: " # N X fi Li P1 . (22) Kð/Þ ¼ L0 þ i¼1
4.3. Procedure for designing the model reference controller To summarize the results of the last two subsections, the following steps can be performed to compute the coefficients matrices G, Q and Mð/Þ and the gain Kð/Þ of the control 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 get ðn mÞ ‘ equations from Eq. (15) and ðn mÞ q equations from Eq. (16). Then, solve these equations for the n q elements of the matrix F. If the matrix F does not exist, then the 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).
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Step 5: Find the vertices Vj ðj ¼ 1; . . . ; 2N Þ, then solve 2N matrix inequalities HðVj Þo0 for the matrices Li (i ¼ 0; 1; . . . ; N), and P. After that, use Eq. (22) to compute the gain Kð/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 1 f1 1 ; B¼ ; C ¼ ½0 1, Að/Þ ¼ 1 0 0 where 1 ¼ f1 f1 :¼ cosðyÞ f1 ¼ 1. Also, consider the reference model (3)–(4) with state-space matrices 2:8 2 2 Að/Þ ¼ ; B¼ ; C ¼ ½0 1. 1 0 0 The following steps are performed to obtain the control law parameters. Step 1: Let " # f1 f2 , F¼ f3 f4 then the matrix G in Eq. (6) is obtained using Eq. (17) such that f1 f2 T T 1 T T 1 G ¼ C ½CC C þ ½I C ½CC CF ¼ , 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 # " U1 S 1 0 1 T ; W ¼ 1, ¼ ¼ U ¼ ; S¼ U2 0 0 1 0 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 satisfied when 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 are calculated from (17)–(19) as follows: 1 0 ; Q ¼ 2; Mð/Þ ¼ ½2:8 f1 1. G¼ 0 1
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1.4 Reference model and system outputs
Output of the reference model Output of the system
1.2 1 0.8 0.6 0.4 0.2 0 0
5
10
15
Time, s Fig. 1. Simulation results for the first model reference control system.
Step 5: The vertices V1 ¼ f1 ¼ 1 and V2 ¼ f1 ¼ 1 are used to solve the following matrix inequalities: HðV1 Þ ¼ Aðf1 ÞP þ B½L0 þ f1 L1 þ %o0, HðV2 Þ ¼ Aðf1 ÞP þ B½L0 þ f1 L1 þ %o0. The following results are obtained using the LMI Control Toolbox [59]: L0 ¼ ½73:7192 0; P¼
L1 ¼ ½81:9103 32:7641,
81:9103 32:7641 . 32:7641 81:9103
Then, Eq. (22) is used to compute the following controller gain: Kð/Þ ¼ ½1:0714 f1 0:4286. The designed model reference controller is applied to the LPV system. Fig. 1 shows the simulation results when the initial conditions are taken to be zero and the reference input is chosen as uðtÞ ¼ 1. It can be seen from Fig. 1 that the system output, yðtÞ, is almost identical to the reference output, yðtÞ. We expect to get this result, because the original dynamic model is used to design the controller. 5.2. Second example In this subsection, the model reference control scheme developed in the previous sections is applied to a coupled-tank process where the process is modeled as an LPV system. The coupled-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 water reservoir to tank 1 and the outflow of tank 1 flows through tank 2 to the water reservoir.
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Inlet Pipe Tank 1
h1
Tank 2
k pu Pump
h2
Water Reservoir
Fig. 2. Schematic diagram of the coupled-tank process.
Pressure sensors located at the bottom of each tank are used to measure the water levels in the tanks. The objective is to design a model reference controller so that the water level in tank 2 tracks the output of a reference model. The dynamics model of the water levels h1 ðtÞ and h2 ðtÞ can be written as [61] a1 pffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi kp 2g h1 ðtÞ þ uðtÞ, h_1 ðtÞ ¼ A1 A1 ffiffiffiffiffiffiffiffiffi ffi p p ffiffiffiffiffi p a1 a2 ffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi 2g h1 ðtÞ 2g h2 ðtÞ, h_2 ðtÞ ¼ A2 A2 yðtÞ ¼ h2 ðtÞ,
(23) (24) (25)
where hi is the water level in tank i; Ai the cross-section area of tank i; ai the cross-section area of tank i outflow orifice; u the voltage applied to pump; kp the gain of pump; g the gravitational constant. The physical quantities are given as follows: A1 ¼ A2 ¼ 15:5179 cm2 ; kp ¼ 3:3 cm3 =V s;
a1 ¼ a2 ¼ 0:1781 cm2 ,
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. pffiffiffiffi First, a standard polynomial fitting technique [62] is used to approximate hi for 0 hi 30 cm with fi hi , where fi ¼ a4 h4i þ a3 h3i þ a2 h2i þ a1 hi þ a0 ,
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with a4 ¼ 2:981 107 ;
a3 ¼ 3:659 105 ;
a1 ¼ 4:036 102 ;
a2 ¼ 1:73 103 ,
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/Þx þ Bu, y ¼ Cx,
(26) (27)
where " x:¼
h1
#
h2
" ;
Að/Þ ¼
0:5085f1 0:5085f1
# 0 ; 0:5085f2
B¼
0:2127 0
;
C ¼ ½0 1.
Remark 4. An LTI reference model (i.e., the reference model (3)–(4) with Ai ¼ 0 for i ¼ 1; 2; . . . ; N) cannot be designed for the coupled-tank system because it can be shown that Eq. (16) reduces to C A0 0:5085f1 F 1 þ 0:5085f2 C ¼ 0,
(28)
where F 1 is the first row of the matrix F. Eq. (28) is satisfied for all possible values of f1 and f2 when C ¼ F 1 ¼ 0 which is unacceptable solution. Therefore, in this case an LPV reference model 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 has been introduced to ensure zero steady-state error. The state-space equations of the LPV reference model can be written in the form (3)–(4) as x_ ¼ Að/Þx þ Bu, y ¼ Cx, with " x¼
# x1 ; x2
Að/Þ ¼
a1 ð/Þ a2 ð/Þ ; 0 bc
B¼
0 ; b
C ¼ ½c 0,
where a1 ð/Þ, a2 ð/Þ, b, and c will be specified later on. The following steps are performed to obtain the control law parameters. Step 1: Let " # f1 f2 , F¼ f3 f4 u
b
( ) dt
x2
x1 = a1( ) x1 +a2 ( ) x2 y = c x1
Fig. 3. Structure of the reference model.
y
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then the matrix G in Eq. (6) is obtained using Eq. (17) such that f1 f2 , G ¼ CT ½CCT 1 C þ ½I CT ½CCT 1 CF ¼ 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 # " U1 S 1 0 0:2127 T ; W ¼ 1, ¼ ¼ U ¼ ; S¼ U2 0 0 1 0 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 ½cða1 ð/Þ þ 0:5085f2 Þ 0:5085f 1 f1 ca2 ð/Þ 0:5085f 2 f1 ¼ 0, which implies that 0:5085f 1 f1 0:5085f2 , c 0:5085f 2 f1 . a2 ð/Þ ¼ c The gains f 1 and f 2 need to be properly designed in order to guarantee the stability of the LPV 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 the following set of LMIs for a positive definite symmetric matrix P [17]: a1 ð/Þ ¼
T
A ðVj ÞP þ P AðVj Þo0
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 LMI Control Toolbox [59] when b ¼ c ¼ 1. Remark 5. If the transient response of the reference model needs to be improved, an additional set of LMIs can be formulated to ensure that the poles of the reference model lie inside a selected sub-region of the left-half plane for all admissible values of fi ; further details can be found in Ref. [18]. Step 4: Using the results in Steps 1–3, the coefficient matrices G, Q and Mð/Þ are calculated from (17)–(19) as follows: 0:01 0:05 ; Q ¼ 0:23507, G¼ 1 0 Mð/Þ ¼ ½0:23507 þ 0:024147f1 0:023907f2 0:12073f1 . Step 5: The following vertices: " # " # f1 f1 0:6 0:6 ; V2 ¼ , ¼ ¼ V1 ¼ f2 f2 0:6 0:1
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" V3 ¼
f1
#
f2
0:1 ¼ ; 0:6
"
V4 ¼
f1 f2
#
0:1 ¼ , 0:1
are used to solve the matrix inequalities: HðV1 Þ ¼ AðV1 ÞP þ B½L0 þ f1 L1 þ f2 L2 þ %o0, HðV2 Þ ¼ AðV2 ÞP þ B½L0 þ f1 L1 þ f2 L2 þ %o0, HðV3 Þ ¼ AðV3 ÞP þ B½L0 þ f1 L1 þ f2 L2 þ %o0, HðV4 Þ ¼ AðV4 ÞP þ B½L0 þ f1 L1 þ f2 L2 þ %o0. The following results are obtained using the LMI Control Toolbox [59]: L0 ¼ ½3:5255 0;
L1 ¼ ½3:5854 4:5817,
L2 ¼ ½0 0:99626;
P¼
1:4997
0:41673
0:41673
1:9394
.
Then, Eq. (22) is used to compute the controller gain as Kð/Þ ¼ ½2:5 þ 1:8444f1 0:1518f2 0:53717 1:966f1 0:5463f2 . The designed model reference controller is applied to the nonlinear model of the coupledtank process. Fig. 4 shows the simulation results when the initial water level in tank 2 is 0 cm. The objective is to force the water level in tank 2 to converge to the desired reference output with a steady state value of 10 cm. It can be seen from Fig. 4 that the water level in tank 2, h2 , follows the desired output of the reference model, y, very closely. Also, the output error e ¼ h2 y is shown in Fig. 5. The result shows that the steady state output error is within 0:2 mm. This nonzero steady state output error might be attributed to the discrepancies between the dynamics model of the process (23)–(25) and the LPV model (26) and (27) which is used to design the controller. 12 Water level in tank 2 Output of the reference model
Water Level in Tank 2, cm
10 8 6 4 2 0 0
50
100
150
200 Time, s
250
300
Fig. 4. Simulation results for the water level in tank 2.
350
400
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12 10
Output Error, cm
8 6 4 2 0 −2 0
50
100
150
200 Time, s
250
300
350
400
Fig. 5. Simulation results for the output error e ¼ h2 y.
Fig. 6. The coupled-tank process.
The designed model reference controller was next implemented on an actual coupledtank process depicted in Fig. 6. The results are presented in Figs. 7 and 8. Fig. 7 shows that the water level in tank 2 follows very closely the desired output of the reference model. Note that the steady state output error is within 0:8 mm (see Fig. 8). The experimental
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12 Water level in tank 2 Output of the reference model
Water Level in Tank 2, cm
10 8 6 4 2 0 0
50
100
150
200
250
300
350
400
Time, s Fig. 7. Experimental results for the water level in tank 2.
12 10
Output Error, cm
8 6 4 2 0 −2 0
50
100
150
200 Time, s
250
300
350
400
Fig. 8. Experimental results for the output error e ¼ h2 y.
steady state output error is larger than the steady state error obtained numerically due to several factors such as: the unmodeled dynamics of the pump, the time delay, the sensor bias, 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
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singular value decomposition of the input matrix, and then obtaining a parameterdependent state feedback gain using linear matrix inequalities. A simple numerical example is used to illustrate the proposed design. To demonstrate the usefulness and the practicality of the proposed control scheme, the model reference control of a coupled-tank process is studied in details. The coupled-tank process is first modeled as an LPV system, then the detailed derivation of the controller is given. The controlled system is simulated and then implemented. The simulation as well as the experimental results indicate that the proposed control scheme works well. Therefore, the proposed scheme can be used to design model reference controllers for LPV systems. Future work will attempt to design robust LPV model reference controllers. References [1] E.W. Kamen, P.P. Khargonekar, On the control of linear systems whose coefficients are functions of parameters, IEEE Transactions on Automatic Control 1 (1984) 25–33. [2] J.S. Shamma, Analysis and design of gain scheduled control systems, Ph.D. Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1988. [3] J.S. Shamma, M. Athans, Guaranteed properties of gain scheduled control for linear parameter-varying plants, Automatica 3 (1991) 559–564. [4] J.S. Shamma, M. Athans, Gain scheduling: potential hazards and possible remedies, IEEE Control Systems Magazine 3 (1992) 101–107. [5] G.I. Bara, J. Daafouz, F. Kratz, J. Ragot, Parameter-dependent state observer design for affine LPV systems, International Journal of Control 16 (2001) 1601–1611. [6] G.I. Bara, J. Daafouz, F. Kratz, Advanced gain scheduling techniques for the design of parameter-dependent observers, in: Proceedings of the IEEE Conference on Decision and Control, 2001, pp. 2822–2827. [7] J.S. Shamma, J.R. Cloutier, Trajectory scheduled missile autopilot design, in: Proceedings of the IEEE Conference on Control Applications, 1992, pp. 237–242. [8] F. Wu, A. Packard, Optimal LQG performance of linear uncertain systems using state-feedback, in: Proceedings of the American Control Conference, 1995, pp. 4435–4439. [9] W. Xie, H 2 gain scheduled state feedback for LPV system with new LMI formulation, IEE ProceedingsControl Theory and Applications 6 (2005) 693–697. [10] C.E. de Souza, A. Trofino, Gain-scheduled H 2 controller synthesis for linear parameter varying systems via parameter-dependent Lyapunov functions, Journal of Robust and Nonlinear Control 5 (2006) 243–257. [11] G.I. Bara, J. Daafouz, Parameter-dependent control with g-performance for affine LPV systems, in: Proceedings of the IEEE Conference on Decision and Control, 2001, pp. 2378–2379. [12] F. Wu, A. Packard, LQG control design for LPV systems, in: Proceedings of the American Control Conference, 1995, pp. 4440–4444. [13] T. Iwasaki, R.E. Skelton, All controllers for the general H 1 control problem: LMI existence conditions and state space formulas, Automatica 30 (1994) 1307–1317. [14] F. Wu, X. Hua Yang, A. Packard, G. Becker, Induced L2 -norm control for LPV system with bounded parameter variation rates, in: Proceedings of the American Control Conference, 1995, pp. 2379–2383. [15] P. Apkarian, P. Gahinet, A convex characterization of gain-scheduled H 1 controllers, IEEE Transactions on Automatic Control 5 (1995) 853–864. [16] P. Apkarian, P. Gahinet, G. Becker, Self-scheduled H 1 control of linear parameter varying systems: a design example, Automatica 9 (1995) 1251–1261. [17] P. Gahinet, P. Apkarian, M. Chilali, Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE Transactions on Automatic Control 3 (1996) 436–442. [18] M. Chilali, P. Gahinet, H 1 design with pole placement constraints: an LMI approach, IEEE Transactions on Automatic Control 3 (1996) 358–367. [19] P. Apkarian, H.D. Tuan, Parameterized LMIs in control theory, SIAM Journal of Control Optimization 4 (2000) 1241–1264. [20] P.J. Oliveira, R.C.L.F. Oliveira, V.J.S. Leite, V.F. Montagner, P.L.D. Peres, H 1 guaranteed cost computation by means of parameter-dependent Lyapunov functions, Automatica 6 (2004) 1053–1061.
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Journal of the Franklin Institute 346 (2009) 854–871 www.elsevier.com/locate/jfranklin
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 parameterdependent 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-space matrices depend on a set of time-varying parameters that are not known in advance, but it can be measured or estimated upon operation of the system. The idea of controlling LPV systems has been introduced in [1–4], then further extended during the last two decades to Corresponding author. Tel.: +965 24987364; fax: +965 24817451.
E-mail address: [email protected] (A. Abdullah). 0016-0032/$32.00 r 2009 The Franklin Institute. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.jfranklin.2009.04.006
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produce many control methods. Examples of these methods include: observer [5,6], state feedback controllers [7], state feedback controllers with LQG performance [8], state feedback controllers with H2 performance [9,10], state feedback controllers with H1 performance [11], output feedback controllers with LQG performance [12], H1 controllers [13–21], output feedback controllers subject to control saturation [22], anti-windup controllers [23,24], model predictive controllers [25,26], adaptive neural network controllers [27], and fuzzy controllers [28,29]. Most of the abovementioned techniques have been applied to practical systems. Control designs 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 in computer-networks and web servers [52,53] have been investigated. The objective 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 proposed LPV model reference control is an extension to the well-known linear time-invariant (LTI) model reference controller that has been extensively studied by many researchers, see for example [54–57] and the references therein. This paper is organized as follows. The second section formulates the problem under investigation. In the following section, the structure of the model reference control is presented using a set of matrix equalities and inequalities which need to be solved for the parameters of the control scheme. The fourth section presents a methodology for the design of the control parameters. In Section 5, the proposed model reference control is applied to a numerical example and to a coupled-tank process; the simulation and experimental results are presented and discussed. Finally, some concluding remarks are given 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: _ ¼ Að/ðtÞÞxðtÞ þ BuðtÞ, xðtÞ
(1)
yðtÞ ¼ CxðtÞ,
(2)
where Að/ðtÞÞ ¼ A0 þ
N X
fi ðtÞAi .
i¼1
The matrices Ai (i ¼ 0; 1; . . . ; N), B and C are known constant matrices of appropriate dimensions. The vector xðtÞ 2 Rn is the state vector, uðtÞ 2 Rm is the input vector and yðtÞ 2 Rp is the output vector. The vector /ðtÞ ¼ ½f1 ðtÞ f2 ðtÞ . . . fN ðtÞT 2 RN is the timevarying 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.
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Assumption 3. The ith element, fi ðtÞ, of the parameter vector /ðtÞ is assumed to vary between known fi and fi , i.e., fi fi ðtÞ 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 online using 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 for example [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 inputs and 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 fi ðtÞ, the augmented model 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 yðtÞ asymptotically, where the reference output yðtÞ is assumed to be obtained from the following LPV reference model: _ ¼ Að/ðtÞÞxðtÞ þ BuðtÞ, xðtÞ
(3)
yðtÞ ¼ CxðtÞ,
(4)
where Að/ðtÞÞ ¼ A0 þ
N X
fi ðtÞAi .
i¼1
The matrices Ai (i ¼ 0; 1; . . . ; N), B and C are known constant matrices of appropriate dimensions. The vector xðtÞ 2 Rq is the reference state vector, uðtÞ 2 R‘ is the reference input 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 of the LPV reference model. Note that in the rest of the paper, the variable dependence on time will be suppressed when no confusion might arise. Also, the symbol % will be used to represent the transpose of a matrix G in the following symmetric matrix: G þ %:¼G þ GT .
3. Model reference control for LPV systems The structure of the model reference control scheme is chosen to be similar to the wellknown structure for the LTI case [56] except that some of the design gain matrices are parameter-dependent.
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Consider the control law given by u ¼ Kð/Þ½x Gx þ Mð/Þx þ Qu,
(5)
where the matrices G, Q, Mð/Þ and Kð/Þ are design matrices of appropriate dimensions satisfying the following relations: C ¼ CG,
(6)
GB ¼ BQ,
(7)
GAð/Þ ¼ BMð/Þ þ Að/ÞG, 1
Kð/Þ ¼ Lð/ÞP ,
(8) (9)
with Að/Þ, B, C, Að/Þ, B, and C are specified in (1)–(2) and (3)–(4). The matrix Lð/Þ and the positive definite symmetric matrix P in Eq. (9) are the solutions of the following matrix inequality: Hð/Þ:¼Að/ÞP þ BLð/Þ þ %o0.
(10)
Remark 3. It should be pointed out that the structure of Lð/Þ has to be chosen in order to synthesize 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 þ
N X
fi Li ,
i¼1
where Li ’s are design constant matrices. The following lemma shows that by using the control law (5), the system output y converges 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/Þx þ Bu GBu ¼ Að/Þx þ BKð/Þ½x Gx þ ½BMð/Þ GAð/Þx þ ½BQ GBu ¼ ½Að/Þ þ BKð/Þe.
(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 eT P1 e.
(12)
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Taking the derivative of Eq. (12) with respect to time and using Eq. (11), the condition _ Wo0 can be written as AT ð/ÞP1 þ P1 Að/Þ þ K T ð/ÞBT P1 þ P1 BKð/Þo0.
(13)
Then pre-multiplying and post-multiplying both sides of inequality (13) by P and using Lð/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 the coefficient matrices G, Q and Mð/Þ as well as the controller gain matrix Kð/Þ. The design of 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 needed for 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 , T
(14)
T
T
T
nm
with U U ¼ UU ¼ I, W W ¼ WW ¼ I, and S 2 R with " # " # U1mn S UT ¼ ; S¼ ; S ¼ diag½s1 s2 . . . sm , U2ðnmÞn 0ðnmÞm where s1 s2 sm 40 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 Rnq satisfying the following relations: U2 GB ¼ 0,
(15)
U2 ½GAð/Þ Að/ÞG ¼ 0,
(16)
where G ¼ CT ½CCT 1 C þ ½I CT ½CCT 1 CF.
(17)
Furthermore, the matrices Q and Mð/Þ are given by Q ¼ WS1 U1 GB, 1
Mð/Þ ¼ WS U1 ½GAð/Þ Að/ÞG.
(18) (19)
Proof. Notice that the general solution of G in Eq. (6) is given by Eq. (17) with F being an arbitrary matrix such that C ¼ CG¼) G ¼ CT ½CCT 1 C þ ½I CT ½CCT 1 CF.
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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 USWT Q¼) UT GB ¼ UT USWT Q() UT GB ¼ SWT Q. Now using
"
T
U ¼
U1
# and
U2
S S¼ , 0
yields the following: " # U1 S WT Q. GB ¼ U2 0 which implies that U1 GB ¼ SWT Q
and
U2 GB ¼ 0.
Note that U1 GB ¼ SWT Q implies that Q ¼ WS1 U1 GB. 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) UT GAð/Þ ¼ UT USWT Mð/Þ þ UT Að/ÞG() UT GAð/Þ ¼ SWT Mð/Þ þ UT Að/ÞG. Now using T
U ¼
"
U1
#
U2
and
S¼
S , 0
yields the following: " # " # U U1 S 1 WT Mð/Þ þ GAð/Þ ¼ Að/ÞG. U2 U2 0 which implies that U1 GAð/Þ ¼ SWT Mð/Þ þ U1 Að/ÞG
(20)
U2 GAð/Þ ¼ U2 Að/ÞG.
(21)
and
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The first result given by Eq. (20) implies that Mð/Þ ¼ WS1 U1 ½GAð/Þ Að/ÞG. The second result given by Eq. (21) implies that U2 ½GAð/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 and Mð/Þ is proved. &4.2. Computation of the gain matrix KðfÞ In order to compute the gain matrix Kð/Þ ¼ Lð/ÞP1 , 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;j Þjvi;j 2 ffi ; fi gg;
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 /, HðVj Þo0 for j ¼ 1; . . . ; 2N . Proof. See [16]. &
Using Lemma 2, the matrix inequality problem (10) can be formulated as a finite matrix inequality problem which can be solved efficiently using an LMI numerical algorithm [58]. Once Li (i ¼ 0; 1; . . . ; N), and P satisfying HðVj Þo0 for j ¼ 1; . . . ; 2N are found, the gain Kð/Þ can be computed using Eq. (9) as follows: " # N X fi Li P1 . (22) Kð/Þ ¼ L0 þ i¼1
4.3. Procedure for designing the model reference controller To summarize the results of the last two subsections, the following steps can be performed to compute the coefficients matrices G, Q and Mð/Þ and the gain Kð/Þ of the control 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 get ðn mÞ ‘ equations from Eq. (15) and ðn mÞ q equations from Eq. (16). Then, solve these equations for the n q elements of the matrix F. If the matrix F does not exist, then the 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).
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Step 5: Find the vertices Vj ðj ¼ 1; . . . ; 2N Þ, then solve 2N matrix inequalities HðVj Þo0 for the matrices Li (i ¼ 0; 1; . . . ; N), and P. After that, use Eq. (22) to compute the gain Kð/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 1 f1 1 ; B¼ ; C ¼ ½0 1, Að/Þ ¼ 1 0 0 where 1 ¼ f1 f1 :¼ cosðyÞ f1 ¼ 1. Also, consider the reference model (3)–(4) with state-space matrices 2:8 2 2 Að/Þ ¼ ; B¼ ; C ¼ ½0 1. 1 0 0 The following steps are performed to obtain the control law parameters. Step 1: Let " # f1 f2 , F¼ f3 f4 then the matrix G in Eq. (6) is obtained using Eq. (17) such that f1 f2 T T 1 T T 1 G ¼ C ½CC C þ ½I C ½CC CF ¼ , 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 # " U1 S 1 0 1 T ; W ¼ 1, ¼ ¼ U ¼ ; S¼ U2 0 0 1 0 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 satisfied when 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 are calculated from (17)–(19) as follows: 1 0 ; Q ¼ 2; Mð/Þ ¼ ½2:8 f1 1. G¼ 0 1
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1.4 Reference model and system outputs
Output of the reference model Output of the system
1.2 1 0.8 0.6 0.4 0.2 0 0
5
10
15
Time, s Fig. 1. Simulation results for the first model reference control system.
Step 5: The vertices V1 ¼ f1 ¼ 1 and V2 ¼ f1 ¼ 1 are used to solve the following matrix inequalities: HðV1 Þ ¼ Aðf1 ÞP þ B½L0 þ f1 L1 þ %o0, HðV2 Þ ¼ Aðf1 ÞP þ B½L0 þ f1 L1 þ %o0. The following results are obtained using the LMI Control Toolbox [59]: L0 ¼ ½73:7192 0; P¼
L1 ¼ ½81:9103 32:7641,
81:9103 32:7641 . 32:7641 81:9103
Then, Eq. (22) is used to compute the following controller gain: Kð/Þ ¼ ½1:0714 f1 0:4286. The designed model reference controller is applied to the LPV system. Fig. 1 shows the simulation results when the initial conditions are taken to be zero and the reference input is chosen as uðtÞ ¼ 1. It can be seen from Fig. 1 that the system output, yðtÞ, is almost identical to the reference output, yðtÞ. We expect to get this result, because the original dynamic model is used to design the controller. 5.2. Second example In this subsection, the model reference control scheme developed in the previous sections is applied to a coupled-tank process where the process is modeled as an LPV system. The coupled-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 water reservoir to tank 1 and the outflow of tank 1 flows through tank 2 to the water reservoir.
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Inlet Pipe Tank 1
h1
Tank 2
k pu Pump
h2
Water Reservoir
Fig. 2. Schematic diagram of the coupled-tank process.
Pressure sensors located at the bottom of each tank are used to measure the water levels in the tanks. The objective is to design a model reference controller so that the water level in tank 2 tracks the output of a reference model. The dynamics model of the water levels h1 ðtÞ and h2 ðtÞ can be written as [61] a1 pffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi kp 2g h1 ðtÞ þ uðtÞ, h_1 ðtÞ ¼ A1 A1 ffiffiffiffiffiffiffiffiffi ffi p p ffiffiffiffiffi p a1 a2 ffiffiffiffiffipffiffiffiffiffiffiffiffiffiffi 2g h1 ðtÞ 2g h2 ðtÞ, h_2 ðtÞ ¼ A2 A2 yðtÞ ¼ h2 ðtÞ,
(23) (24) (25)
where hi is the water level in tank i; Ai the cross-section area of tank i; ai the cross-section area of tank i outflow orifice; u the voltage applied to pump; kp the gain of pump; g the gravitational constant. The physical quantities are given as follows: A1 ¼ A2 ¼ 15:5179 cm2 ; kp ¼ 3:3 cm3 =V s;
a1 ¼ a2 ¼ 0:1781 cm2 ,
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. pffiffiffiffi First, a standard polynomial fitting technique [62] is used to approximate hi for 0 hi 30 cm with fi hi , where fi ¼ a4 h4i þ a3 h3i þ a2 h2i þ a1 hi þ a0 ,
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with a4 ¼ 2:981 107 ;
a3 ¼ 3:659 105 ;
a1 ¼ 4:036 102 ;
a2 ¼ 1:73 103 ,
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/Þx þ Bu, y ¼ Cx,
(26) (27)
where " x:¼
h1
#
h2
" ;
Að/Þ ¼
0:5085f1 0:5085f1
# 0 ; 0:5085f2
B¼
0:2127 0
;
C ¼ ½0 1.
Remark 4. An LTI reference model (i.e., the reference model (3)–(4) with Ai ¼ 0 for i ¼ 1; 2; . . . ; N) cannot be designed for the coupled-tank system because it can be shown that Eq. (16) reduces to C A0 0:5085f1 F 1 þ 0:5085f2 C ¼ 0,
(28)
where F 1 is the first row of the matrix F. Eq. (28) is satisfied for all possible values of f1 and f2 when C ¼ F 1 ¼ 0 which is unacceptable solution. Therefore, in this case an LPV reference model 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 has been introduced to ensure zero steady-state error. The state-space equations of the LPV reference model can be written in the form (3)–(4) as x_ ¼ Að/Þx þ Bu, y ¼ Cx, with " x¼
# x1 ; x2
Að/Þ ¼
a1 ð/Þ a2 ð/Þ ; 0 bc
B¼
0 ; b
C ¼ ½c 0,
where a1 ð/Þ, a2 ð/Þ, b, and c will be specified later on. The following steps are performed to obtain the control law parameters. Step 1: Let " # f1 f2 , F¼ f3 f4 u
b
( ) dt
x2
x1 = a1( ) x1 +a2 ( ) x2 y = c x1
Fig. 3. Structure of the reference model.
y
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then the matrix G in Eq. (6) is obtained using Eq. (17) such that f1 f2 , G ¼ CT ½CCT 1 C þ ½I CT ½CCT 1 CF ¼ 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 # " U1 S 1 0 0:2127 T ; W ¼ 1, ¼ ¼ U ¼ ; S¼ U2 0 0 1 0 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 ½cða1 ð/Þ þ 0:5085f2 Þ 0:5085f 1 f1 ca2 ð/Þ 0:5085f 2 f1 ¼ 0, which implies that 0:5085f 1 f1 0:5085f2 , c 0:5085f 2 f1 . a2 ð/Þ ¼ c The gains f 1 and f 2 need to be properly designed in order to guarantee the stability of the LPV 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 the following set of LMIs for a positive definite symmetric matrix P [17]: a1 ð/Þ ¼
T
A ðVj ÞP þ P AðVj Þo0
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 LMI Control Toolbox [59] when b ¼ c ¼ 1. Remark 5. If the transient response of the reference model needs to be improved, an additional set of LMIs can be formulated to ensure that the poles of the reference model lie inside a selected sub-region of the left-half plane for all admissible values of fi ; further details can be found in Ref. [18]. Step 4: Using the results in Steps 1–3, the coefficient matrices G, Q and Mð/Þ are calculated from (17)–(19) as follows: 0:01 0:05 ; Q ¼ 0:23507, G¼ 1 0 Mð/Þ ¼ ½0:23507 þ 0:024147f1 0:023907f2 0:12073f1 . Step 5: The following vertices: " # " # f1 f1 0:6 0:6 ; V2 ¼ , ¼ ¼ V1 ¼ f2 f2 0:6 0:1
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" V3 ¼
f1
#
f2
0:1 ¼ ; 0:6
"
V4 ¼
f1 f2
#
0:1 ¼ , 0:1
are used to solve the matrix inequalities: HðV1 Þ ¼ AðV1 ÞP þ B½L0 þ f1 L1 þ f2 L2 þ %o0, HðV2 Þ ¼ AðV2 ÞP þ B½L0 þ f1 L1 þ f2 L2 þ %o0, HðV3 Þ ¼ AðV3 ÞP þ B½L0 þ f1 L1 þ f2 L2 þ %o0, HðV4 Þ ¼ AðV4 ÞP þ B½L0 þ f1 L1 þ f2 L2 þ %o0. The following results are obtained using the LMI Control Toolbox [59]: L0 ¼ ½3:5255 0;
L1 ¼ ½3:5854 4:5817,
L2 ¼ ½0 0:99626;
P¼
1:4997
0:41673
0:41673
1:9394
.
Then, Eq. (22) is used to compute the controller gain as Kð/Þ ¼ ½2:5 þ 1:8444f1 0:1518f2 0:53717 1:966f1 0:5463f2 . The designed model reference controller is applied to the nonlinear model of the coupledtank process. Fig. 4 shows the simulation results when the initial water level in tank 2 is 0 cm. The objective is to force the water level in tank 2 to converge to the desired reference output with a steady state value of 10 cm. It can be seen from Fig. 4 that the water level in tank 2, h2 , follows the desired output of the reference model, y, very closely. Also, the output error e ¼ h2 y is shown in Fig. 5. The result shows that the steady state output error is within 0:2 mm. This nonzero steady state output error might be attributed to the discrepancies between the dynamics model of the process (23)–(25) and the LPV model (26) and (27) which is used to design the controller. 12 Water level in tank 2 Output of the reference model
Water Level in Tank 2, cm
10 8 6 4 2 0 0
50
100
150
200 Time, s
250
300
Fig. 4. Simulation results for the water level in tank 2.
350
400
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12 10
Output Error, cm
8 6 4 2 0 −2 0
50
100
150
200 Time, s
250
300
350
400
Fig. 5. Simulation results for the output error e ¼ h2 y.
Fig. 6. The coupled-tank process.
The designed model reference controller was next implemented on an actual coupledtank process depicted in Fig. 6. The results are presented in Figs. 7 and 8. Fig. 7 shows that the water level in tank 2 follows very closely the desired output of the reference model. Note that the steady state output error is within 0:8 mm (see Fig. 8). The experimental
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12 Water level in tank 2 Output of the reference model
Water Level in Tank 2, cm
10 8 6 4 2 0 0
50
100
150
200
250
300
350
400
Time, s Fig. 7. Experimental results for the water level in tank 2.
12 10
Output Error, cm
8 6 4 2 0 −2 0
50
100
150
200 Time, s
250
300
350
400
Fig. 8. Experimental results for the output error e ¼ h2 y.
steady state output error is larger than the steady state error obtained numerically due to several factors such as: the unmodeled dynamics of the pump, the time delay, the sensor bias, 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
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singular value decomposition of the input matrix, and then obtaining a parameterdependent state feedback gain using linear matrix inequalities. A simple numerical example is used to illustrate the proposed design. To demonstrate the usefulness and the practicality of the proposed control scheme, the model reference control of a coupled-tank process is studied in details. The coupled-tank process is first modeled as an LPV system, then the detailed derivation of the controller is given. The controlled system is simulated and then implemented. The simulation as well as the experimental results indicate that the proposed control scheme works well. Therefore, the proposed scheme can be used to design model reference controllers for LPV systems. Future work will attempt to design robust LPV model reference controllers. References [1] E.W. Kamen, P.P. Khargonekar, On the control of linear systems whose coefficients are functions of parameters, IEEE Transactions on Automatic Control 1 (1984) 25–33. [2] J.S. Shamma, Analysis and design of gain scheduled control systems, Ph.D. Thesis, Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 1988. [3] J.S. Shamma, M. Athans, Guaranteed properties of gain scheduled control for linear parameter-varying plants, Automatica 3 (1991) 559–564. [4] J.S. Shamma, M. Athans, Gain scheduling: potential hazards and possible remedies, IEEE Control Systems Magazine 3 (1992) 101–107. [5] G.I. Bara, J. Daafouz, F. Kratz, J. Ragot, Parameter-dependent state observer design for affine LPV systems, International Journal of Control 16 (2001) 1601–1611. [6] G.I. Bara, J. Daafouz, F. Kratz, Advanced gain scheduling techniques for the design of parameter-dependent observers, in: Proceedings of the IEEE Conference on Decision and Control, 2001, pp. 2822–2827. [7] J.S. Shamma, J.R. Cloutier, Trajectory scheduled missile autopilot design, in: Proceedings of the IEEE Conference on Control Applications, 1992, pp. 237–242. [8] F. Wu, A. Packard, Optimal LQG performance of linear uncertain systems using state-feedback, in: Proceedings of the American Control Conference, 1995, pp. 4435–4439. [9] W. Xie, H 2 gain scheduled state feedback for LPV system with new LMI formulation, IEE ProceedingsControl Theory and Applications 6 (2005) 693–697. [10] C.E. de Souza, A. Trofino, Gain-scheduled H 2 controller synthesis for linear parameter varying systems via parameter-dependent Lyapunov functions, Journal of Robust and Nonlinear Control 5 (2006) 243–257. [11] G.I. Bara, J. Daafouz, Parameter-dependent control with g-performance for affine LPV systems, in: Proceedings of the IEEE Conference on Decision and Control, 2001, pp. 2378–2379. [12] F. Wu, A. Packard, LQG control design for LPV systems, in: Proceedings of the American Control Conference, 1995, pp. 4440–4444. [13] T. Iwasaki, R.E. Skelton, All controllers for the general H 1 control problem: LMI existence conditions and state space formulas, Automatica 30 (1994) 1307–1317. [14] F. Wu, X. Hua Yang, A. Packard, G. Becker, Induced L2 -norm control for LPV system with bounded parameter variation rates, in: Proceedings of the American Control Conference, 1995, pp. 2379–2383. [15] P. Apkarian, P. Gahinet, A convex characterization of gain-scheduled H 1 controllers, IEEE Transactions on Automatic Control 5 (1995) 853–864. [16] P. Apkarian, P. Gahinet, G. Becker, Self-scheduled H 1 control of linear parameter varying systems: a design example, Automatica 9 (1995) 1251–1261. [17] P. Gahinet, P. Apkarian, M. Chilali, Affine parameter-dependent Lyapunov functions and real parametric uncertainty, IEEE Transactions on Automatic Control 3 (1996) 436–442. [18] M. Chilali, P. Gahinet, H 1 design with pole placement constraints: an LMI approach, IEEE Transactions on Automatic Control 3 (1996) 358–367. [19] P. Apkarian, H.D. Tuan, Parameterized LMIs in control theory, SIAM Journal of Control Optimization 4 (2000) 1241–1264. [20] P.J. Oliveira, R.C.L.F. Oliveira, V.J.S. Leite, V.F. Montagner, P.L.D. Peres, H 1 guaranteed cost computation by means of parameter-dependent Lyapunov functions, Automatica 6 (2004) 1053–1061.
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