6.1. INTRODUCTION
Multi-input multi-output (MIMO) systems consist of many process variables including measurement and control signals with complicated interactions that make it difficult to control. The well-known quadruple- tank process, proposed by Johansson [1], is a common system for evaluating control strategies of multivariable processes. Many researchers have developed control algorithms for improving their performances in terms of servomechanism problems and disturbance rejection [2–7]. However, from the conventional controllers to the advanced control algorithms, the process delays are neglected in the system model. In real applications, they are responsible for the degradation of system performance at higher values of gain [7].
In this work, the simplified decoupling method proposed by Vu et al. [8] is adopted to deal with interactions between process variables.
To overcome the realizability problem of the decoupling techniques and enhance dynamic behaviors of decoupled systems, fractional-order processes are suggested to be the equivalent transfer functions of the decoupled elements. The particle swarm optimization (PSO) algorithm for the approximation procedure proposed in [9] is employed to find out the parameters of approximated fractional functions. Bouyedda et al. [10]
performed a similar work by using the Genetic Algorithm (GA) to reduce a high integer-order transfer function of a SISO system to a lower fractional- order one.
Fractional calculus is a mathematical phenomenon that helps to describe dynamic behaviors of real plants more accurately than the integer-order conventional methods. In the control field, fractional calculus had been widely used since Oustaloup introduced an approximation approach of
fractional derivative and integral in the frequency domain [11]. Therefore, in recent years, the fractional-order proportional-integral-derivative (FOPID) controller has attracted more attention from many researchers.
The FOPID has five tuning parameters including proportional, integral, derivative gain, and fractional orders of the integral and derivative terms which provide more flexibility in system performances as well as robustness compared with the conventional one [12–14]. Because of more tuning parameters, it is also harder to derive analytical tuning rules for the controller. Different tuning methods have been suggested to solve this kind of problem [15–20]. However, most of them are used to deal with single- input single-output (SISO) systems.
In this paper, the FOPI based IMC is presented to find out analytical tuning rules of the controllers for MIMO processes and it is also validated on the quadruple-tank system. The time delays of the system are also considered in the design procedure by using the Padé approximation technique for exponential terms [25]. The proposed method uses the IMC scheme to reduce the number of tuning parameters; and normally, there is only one parameter left that needs to be tuned based on some criteria of system performances in terms of set-point tracking as well as disturbance rejection [10, 18–20]. The identification of process parameters plays a crucial role in multivariable control design for industrial applications because a better performance will be achieved by model-based tuning algorithms. In this work, the least-squared method which is a reliable technique for system identification is used for MIMO systems. The diagonal form of matrix fraction description is suggested to transform a MIMO system into several multi-input single-output sub-models [21-23].
This chapter is organized as follows. Section 6.2 is briefly introduced fractional-order calculus with the Oustaloup recursive algorithm to approximate the fractional operator. Section 6.3 will discuss the Quadruple tank system including its dynamics and the approach used for system identification. The controller structure is mentioned in section 6.4; and a new fractional-order PI controller based on the IMC structure is also
proposed and its analytical tuning rules are derived in this section. The obtained controller will be justified on the real model and the results are included in section 6.5. Finally, conclusions are given in section 6.6.
6.2. PRELIMINARY
Fractional order calculus
Fractional calculus is a generalization of ordinary calculus by extending the integration and differentiation order to the non-integer order. It presented a fractional operatoraDtv where a and t are the limits and v is the fractional order (v R∈ ). The most common definition of the fractional operator was proposed by Riemann and Liouville [13] and it is defined as following
1
1 ( ) , 1
( ) ( )
v t
a t n v n
a
d f
D t dt n v n
n v dt t τ τ − +
= − < <
Γ − ∫ − (6.1)
where Γ •( ) represents the Euler’s gamma function; with a positive v, the fractional operator denotes fractional derivative, and a negative v represents fractional integral.
The fractional-order transfer function of a SISO system can be described:
1 0
1 0
1 0
1 0
( ) m vmn m vnm v
n n
b s b s b s
G s a s a s a s
λ λ− λ
−
−
−
+ + +
= + + +
(6.2)
The fractional-order of s in equation (6.2) makes it difficult to simulate or implement a fractional-order system. Therefore, Oustaloup proposed an approximation method using a recursive algorithm with finite numbers of poles and zeros for most applications [11, 13]. Within the specific range of frequency[ω ωb, h], the approximation of the FO operator, sv, can be obtained by the following equation:
[ b, h]
v v N z
k N p
s s K s
ω ω s
ω
=− ω
≅ +
∑ +
(6.3)
where the zero, pole and gain can be calculated respectively from:
hv
K =ω (6.4)
(k N 0.5 0.5 ) (2v N 1)
z b h
b
ω ω ω ω
+ + − +
=
(6.5)
(k N 0.5 0.5 ) (2v N 1)
p b h
b
ω ω ω ω
+ + + +
=
(6.6)
6.3. QUADRUPLE-TANK SYSTEM
6.3.1. The system description and its dynamics
Fig. 6.1 shows the pipe and instrumentation diagram (P&ID) of the quadruple-tank system. The inlet liquids are pumped by two centrifugal pumps into four tanks with cross-connection as shown in the figure.
Manipulated variables are two analog voltages (u1, u2:0÷10 VDC) which are applied to the inverters to control the power of the two pumps and then manipulate the flow rates of the inlet streams. The two three-way valves (V1, V2) with the adjustable coefficientsγ γ1, 2 (∈[ ]0,1 ) determine the portion of their outputs to the upper and lower tanks respectively. The outputs of this system are two levels of the lower tanks (h1, h2) which are measured by level sensors, LR2750 of IFM (LT1, LT2).
q3 q4
q3o q4o
V2
Inv1
P1
Inv2 u2
P2
FT1 FT2
V1
h1
h2
LT1 LT2
h3
h4
u1
q1
q2
q1o q2o
1 2
FT ,FT : Flow Transmitters
1 2
LT ,LT : Level Transmitters
1 2
P ,P : Centrifugal pumps
1 2
V ,V : Three-way valves Inv1,Inv2 : Inverters
Figure 6.1. The piping and instrumentation diagram of the quadruple-tank system
Applying the mass conservation law and Bernoulli’s law, the dynamic equations of the system are available in some literature [1, 4, 6, 7]. In this work, they are linearized around the operating points: hi0 corresponding to
the inputs (u u10, 20). And let define new variables:
0 ( 1 4)
i i i
x h h i= − = ÷ (6.7)
0 ( 1 2)
j j j
r u u= − j= ÷ (6.8)
Then, the linearization equations are obtained as following:
3
1 2 2 2
1 3
1 1 3 1
1 A
dx x x k r
dt A A
γ
τ τ
= − + + (6.9)
2 4 1 1 1
2 4
2 2 4 2
1
dx x A x k r
dt A A
γ
τ τ
= − + + (6.10)
3 1 1 1
3
3 3
(1 )
dx 1 x k r
dt A
γ τ
= − + − (6.11)
4 2 2 2
4
4 4
(1 ) 1
dx x k r
dt A
γ τ
= − + − (6.12)
where i i 2 i0 ( 1 4)
i
A h i
a g
τ = = ÷ : time constant (s) (6.13)
Aiis the cross-sectional area of the tank i, aiis the cross-sectional area of the outlet hole of the tank i; it is assumed that the characteristics of the pumps are linear. So, the relationship between the applied voltage and the outlet flow rate of each pump is a constant represented by ki; it is also ignored the delay time for the liquid that travels from the pump to the tanks.
Using Laplace transform to convert the differential equations into the matrix form of transfer functions, we obtain:
1 1 2 2
1 3 1
1 1
2 3 1 2 2 2
2 2 4
(1 )
( 1)( 1) 1
( ) ( )
( ) (1 ) ( )
1 ( 1)( 1)
K K
s s s
X s R s
X s K K R s
s s s
γ γ
τ τ τ
γ γ
τ τ τ
−
+ + +
=
−
+ + +
(6.14)
where 1 1 1 2 1 2 3 2 1 4 2 2
1 2 2 2
; ; ;
k k k k
K K K K
A A A A
τ τ τ τ
= = = =
Note that, in Eq. (6.14), for its simplicity, the time delays are ignored.
In real-time application, however, the liquid needs time to travel from the
pumps to the tanks. Therefore, each transfer function in Eq. (6.14) will be in series with a time delay term.
6.3.2. Identification of the quadruple-tank system
The theoretical model of the quadruple-tank system is derived in the previous section. It is shown that the system nonlinear model could be approximated to the linear form of the two-input, two-output (TITO) system.
In this part, the least-squares method will be adopted to identify each transfer function of the whole system. The general linear form can be described by:
1 11 12 1 1
2 21 22 2 2
( ) ( ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( )
y t G q G q u t v t
y t G q G q u t v t
= +
(6.15)
where G qij( ) is a rational polynomial in the delay operator
In this paper, the diagonal form of matrix fraction description (MFD) is used due to its simplicity and applicability. As a result of that, the TITO process is decoupled into two two-input single-output sub-models which are identified separately. The diagonal form MFD is given as
1
11 12 1 11 12
21 22 2 21 22
( ) ( ) ( ) 0 ( ) ( )
( ) ( ) ( ) 0 ( ) ( ) ( )
G q G q A q B q B q
G q G q G q A q B q B q
−
= =
(6.16)
whereA qi( )and B qij( ) (i j, = ÷1 2) are polynomials in the delay operator q−1:
1 2
,1 ,2 ,
( ) 1 i
i
i i i i na na
A q = +a q− +a q− + + a q− (6.17)
1 1
1 2
1j( ) 1 ,1j 1 ,2j 1 ,j nbj nbj
B q =b q− +b q− + + b q− (6.18)
2 2
1 2
2j( ) 2 ,1j 2 ,2j 2 ,j nb j nbj
B q =b q− +b q− + + b q− (6.19)
Replace (6.16) into (6.15), it yields:
1 1 11 1 12 2 1 1
2 2 21 1 22 2 2 2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A q y t B q u t B q u t A q v t A q y t B q u t B q u t A q v t
= + +
= + +
(6.20)
Considering the first sub-model, the least-squares estimation of the system parameters to minimize the loss function:
[ ]
( )
1
2
1 1 1 11 1 12 2
1
( ) ( ) ( ) ( ) ( ) ( )
N
LS t na
V A q y t B q u t B q u t
= +
= ∑ − + (6.21)
The solution is obtained in the following form [21, 23]
(6.22) where
1 1
1 1
1
1
( 1)
( 2)
( ) y na y na y N
+
+
=
y
Using the PRBS signal to excite the system around its initial operating point, then the pair of input-output data are collected. The sampling time is chosen as Ts =0.1( )s and after the pretreatment of the data, 2000 samples are available. The first 1200 samples are used for model identification and the rest of data (800 samples) are used for validation. Fig. (6.2) and (6.3) prove that the identified models are quite accurate, and therefore, can be used for control purposes.
0 100 200 300 400 500 600 700 800
Sample -6
-4 -2 0 2 4 6
identified validation data
Figure 6.2. The validation of identified sub-model 1
0 100 200 300 400 500 600 700 800
Sample -10
-5 0 5 10 15
identified validation data
Figure 6.3. The validation of identified sub-model 2
In both sub-models, usingnai =2, 2, 2,nb1j = nb2j = the model parameters are obtained as follows:
[ ]
1= −1.3692 0.3725 0.0743 0.0737 T
ử (6.23)
[ ]
2 = −1.2289 0.2305 0.0023 0.0722T
ử (6.24)
Convert into the continuous transfer functions with the specified sampling time. The elements of the matrix transfer function of the system are derived and approximated by first-order plus time-delay systems. The results are summarized in Table 6.1.
Table 6.1. The identified transfer functions and their approximation Identified transfer functions Approximated
11 2
0.8229 15.08 ( ) 12.46 0.6891 G s s
s s
− +
= + +
0.1 11( ) 21.885
18.0364 1 e s
G s s
= −
+
12 2
0.7136 15.2 ( ) 12.46 0.6891 G s s
s s
− +
= + +
0.8 12( ) 22.0306
17.2757 1 e s
G s s
= −
+
21 2
0.2244 0.7376 ( ) 20.53 0.4869 G s s
s s
− +
= + +
2 21( ) 1.5101
39.9664 1 e s
G s s
= −
+
22 2
1.694 20.13 ( ) 20.53 0.4869 G s s
s s
− +
= + +
1.7 21( ) 41.2358
40.2471 1 e s
G s s
= −
+
6.4. SIMPLIFIED DECOUPLING BASED ON FRACTIONAL ORDER SYSTEMS
In this work, the simplified decoupling is used to deal with the main issue of multivariable processes, the interactions between process variables.
The control structure is shown in Fig. 6.4 where G(s), D(s), Q(s) are the transfer function matrix, the decoupling matrix, and the decoupled matrix of a TITO process, respectively. Due to the properties of the simplified decoupling [8, 9], they have the forms as follows:
11 12 12 11
21 22 21 22
( ) ( ) 1 ( ) ( ) 0
( ) ; ( ) ; ( )
( ) ( ) ( ) 1 0 ( )
G s G s D s Q s
s s s
G s G s D s Q s
= = =
G D Q (6.25)
where the elements of D(s) and Q(s) can be calculated as follows [8, 9]:
12 12
11
( ) 1.0067(18.0364 1)
( ) ( ) 17.2757 1
G s s
D s G s s
= − = − +
+ (6.26)
21 21
22
( ) 0.0366(40.2471 1)
( ) ( ) 39.9664 1
G s s
D s G s s
= − = − +
+ (6.27)
0.8 2
12 21 0.1
11 11 1.7
22
22.0306 1.5101 ( ) ( ) 21.885 (17.2757 1)(39.9664 1) ( ) ( )
41.2358 ( ) 18.0364 1
40.2471 1
s s
s
s
e e
G s G s e s s
Q s G s
e
G s s
s
− −
−
−
+ +
= − = −
+
+
(6.28)
0.8 2
12 21 1.7
22 22 0.1
11
22.0306 1.5101 ( ) ( ) 41.2358 (17.2757 1)(39.9664 1) ( ) ( )
21.885 ( ) 40.2471 1
18.0364 1
s s
s
s
e e
G s G s e s s
Q s G s
e
G s s
s
− −
−
−
+ +
= − = −
+
+
(6.29)
r1
r2
1( ) G sc
2( ) G sc
y1
y2 12( )
D s
21( ) D s
11( ) G s
12( ) G s
21( ) G s
22( ) G s ( )s Q ( )s
Gc
Figure 6.4. The simplified decoupling structure with fractional control The diagonal elements of the decoupled matrix are complicated as shown in Eq. (6.28) - (6.29). Therefore, various methods were proposed to approximate them to some standard forms. However, most of them only deal with integer-order transfer functions. In this paper, a fractional-order transfer function (FOTF) is proposed to be the equivalent transfer function of Eq. (6.28) and (6.29). The general form of the FOTF is as follows:
2 1 1 2
2 1
( ) (0 1 2)
1
s
m Ke
G s s s
θ
α α α α
τ τ
= − < ≤ < <
+ + (6.30)
whereτ τ1, 2 are time constants; K is a gain; α α1, 2 are fractional orders;
θ is a delay time. In a special case, when τ2=0 Eq. (6.30) becomes the fractional first order transfer function.
The PSO algorithm for approximation proposed by Chuong and et al.
[9] is adopted and expanded for the fractional-order case. Note that the parameter θ is a prior value determined by the unit step response of the original model. Consequently, the number of tuning parameters, in this case, has the following vector form:
[ 2 1 2 1]
x= K τ τ α α (6.31)
The different constraints of these parameters are given in (6.32):
min max
1 max
2 max
1 2
0 0
0 1
1 2
K K K
τ τ τ τ
α α
< <
< <
≤ <
< ≤
< <
(6.32)
whereKmin, , Kmax τmax are determined based on the open-loop unit step response of the original model.
The results of the algorithm are shown in Eq. (6.33). Then, the fractional PI controller will be designed for each loop of the decoupled system.
1.5
11 0.9523
21.3233 ( ) 15.8183 1
e s
Q s s
= −
+
3
22 0.9728
39.9728 ( ) 40.2964 1
e s
Q s s
= −
+ (6.33) 6.5. FRACTIONAL IMC-PI CONTROLLER DESIGN
In this study, a new structure of a fractional PI controller is proposed for each loop, called IσPIλ, and a lead/lag filter is also employed to improve the output performances. Consequently, the primary controller of each loop is as Eq. (6.34)
1 1
( ) 1 ( )
c c
I
G s K F s
sσ τ sλ
= +
(6.34)
where Kc and τI are proportional gain and integral time respectively;
λ is fractional order of the integral term; σ is the fractional order of the ideal integral and σ = −1 λ; in special case, whenλ=1 (integral term
with integer order) then σ equals to zero (it becomes a conventional PI controller); F(s) is the lead-lag filter as Eq. (6.35)
( ) 1
a 1
b
F s s s τ τ
= +
+ (6.35)
where τ τa, b are time constants
The IMC-based PID procedure normally used for integer-order processes is also addressed to design the proposed controller for fractional- order processes. Fig. 6.5 (a) and (b) show block diagrams of feedback control strategies including the classical feedback control and the internal model control as well [9, 18-20].
( )
G sc Q s( ) d
r u y
(a)
c( )
G s Q s( )
r u y
( ) Q s
d
(b)
Figure 6.5. The one degree of freedom feedback control diagram (a). Classical feedback control (b). Internal model control
In this work, the decoupled process model has the fractional first order form after approximating:
( ) (0 1) 1
Ke s
Q s s
θ
α α
τ
= − < <
+ (6.36)
The conventional controller is derived based on IMC structure as follows:
( ) 1
( 1)
c s
c
G s s
K s e
α θ
τ
τ −
= +
+ −
(6.37)
where τc is an adjustable parameter which controls the tradeoff between the performance and robustness.
It is necessary to handle the time delay term in Eq. (6.37) properly to convert Gc into a familiar form of controller in industrial applications. In this paper, a Padé 1/1 approximation is used [25]:
1 0.5 1 0.5
s s
e s
θ θ
θ
− = −
+ (6.38)
The ideal feedback controller is obtained by:
[ ( 1)(1 0.5 ) ] ([ 1)(1 0.5 ) ]
( ) ( 1)(1 0.5 ) (1 0.5 ) 0.5 ( )
c
c c c
s s s s
G s K s s s Ks s
α α
τ θ τ θ
τ θ θ τ θ τ θ
+ + + +
= =
+ + − − + + (6.39)
Rewritten Eq. (6.39) into the form of Eq. (6.34), the controller is derived as Eq. (6.40):
1
1 1 (1 0.5 )
( ) ( ) 1 0.5 1
c c c
c
G s s
K s α sα s
τ θ
τ θ τ τ θ
τ θ
−
+
= + + +
+
(6.40)
( ); ;
c I
c
K K
τ τ τ λ α
= τ θ = =
+
1 ; 0.5 ; a b 0.5 c
c
σ α τ θ τ τ θ
= − = = τ θ
+ 6.6. EXPERIMENTAL RESULTS
The realization of fractional control and the simplified decoupling technique is justified by an experiment on the real model. The obtained decoupler (Eq. 6.36 and 6.27) and the two diagonal elements of the decoupled matrix (Eq. 6.33) are used for controller design. The proposed fractional PI controllers are derived based on Eq. (6.40) in which τc is chosen as 2 and 5 for each control loop respectively. The results are shown as follows:
1 0.0477
1 1 (0.75 1)
( ) 0.212 1
15.8183 0.4286 1
c s
G s s s s
+
= + + (6.41)
2 0.0272
1 1 (1.5 1)
( ) 0.126 1
40.2964 0.9375 1
c s
G s s s s
+
= + + (6.42)
The control structure is implemented by using Simulink of Matlab in Real-Time Window Target based on a supported PCI card, 6323e of National Instrument. The time responses of the levels in both tanks are shown in Fig. 6.6 and Fig. 6.7.
0 20 40 60 80 100 120 140 160 180 200 Time (s)
0 2 4 6 8 10 12 14 16
h1(cm)
Setpoint = 15
Figure 6.6. The step response of the level in tank1 with setpoint 15 (cm)
0 20 40 60 80 100 120 140 160 180 200
Time (s) 0
2 4 6 8 10 12
h2 (cm)
Setpoint = 10
Figure 6.7. The step response of the level in tank2 with setpoint 10 (cm) From the figures, it is obvious that the responses in both tanks are quite good. It is a fast response with no overshoot in tank 1. In tank 2, there is an overshoot of 10% and therefore, it makes the settling time is slower compared to that of tank 1. The steady-state errors of both tanks are about zero.
6.7. CONCLUSION
In this paper, an approach to control a MIMO process is proposed by adopting the simplified decoupling technique and fractional PI controller.
The PSO algorithm is also used to approximate the complex elements of the decoupled matrix to the fractional-order transfer function. The analytical tuning rules of the proposed fractional PI controller are derived based on the IMC structure. The tuning parameter is only a time constant to compromise the tradeoff between the system performance and its robustness. The Quadruple tank system is chosen to justify the effectiveness of the proposed method. A well-known identification algorithm, the least- squared method, is applied for a MIMO system to obtain the linearized model of the tank system. The experimental results prove the effectiveness of the proposed control structure in terms of setpoint tracking.
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Chapter 7