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Adaptive and Non-Adaptive Dynamic Matrix Control for Conical Tank
Adaptive and Non-Adaptive Dynamic Matrix Control for Conical Tank W. Klopot, T. Klopot, M. Metzger Faculty of Automatic Control, Electronics and Computer Science Silesian University of Technology, Akademicka 16, 44-100 Gliwice, Poland, { witold.klopot, tomasz.klopot, mieczyslaw.metzger}@polsl.pl Abstract: Conical tanks are widely used in food process industries, hydrometallurgical industries and also in the water treatment industry. Moreover, liquid level control of a conical tank is still challenging for typical process control because of occurred nonlinearities. This paper presents comparison between Adaptive and Non-Adaptive version of popular predictive Dynamic Matrix Control.
1. INTRODUCTION Control of flow and liquid level is a common and important task in process industries. There are many examples where such a control is used e.g.: petrochemical industries, paper making industries, water treatment industries etc. In each step of a process there can be found tanks which size and shape is dictated by process technology requirements e.g. for corrosive liquids could be used a shape of sphere where lateral area is the smallest in comparison to volume. It is important when is taken into consideration an expense of construction. Besides sphere shape can stand greater tension than other shapes. The other example which impose a great requirements for process control can be a conical tank. It is widely used in industries (Bhuvaneswari et al., 2008) e.g. food process industries, hydrometallurgical industries and especially in water treatment industries where it improves disposal of solids while mixing. This shape is a great challenge for process control. Conical tank will be considered in this paper. It was especially chosen because of its nonlinearities. This nonlinearities causing that plant parameters change more than two ranks. Obviously, the advanced PID controller could be used, but in spite of its common known advantages the quality of control could be not satisfied. Therefore, it was decided to choose Dynamic Matrix Control (DMC) algorithm (Cutler and Ramaker, 1980; Cooper, 2004) and show its comparison between adaptive and non-adaptive version. That kind of choice was done because in some special cases traditional algorithms do not deal with difficult processes and then raise a necessary of using advanced algorithm control. Among all advanced algorithms the predictive control is the most frequently used (Qin and Badgwell, 2003), where DMC algorithms enjoy a great popularity. Predictive control has many advantages nevertheless its number of practical implementation is still smaller than the number of practical applications of conventional controllers PID. Conventional controllers are still playing a main role in control applications (Aström and Hagglund, 1996; Vrancic et al., 2004). It is caused by more complicated construction of an advanced algorithm and also
by difficulties of controller tuning. We should take into account that each attempt which can emphasize economical gain is worth doing. That is why we decided to choose this algorithm. In this work we will base on “Tuning sequence of the DMC controller” which was proposed by (Kłopot and Metzger, 2005; Kłopot et al., 2007). Simulation studies and real-time DMC are discussed. 2. CONTROL PLANT DESCRIPTION
Controlled system is simulated (Metzger, 2000) in LabView environment which provides real-time simulation. The conical tank (Fig.1.) is a part of real laboratory installation (Metzger, 2007) (Fig.2.). It is assumed process value (PV) would be a level liquid and would be maintained by changing the input flow – control value (CV). Operating point is set by valve.
Fig.1. Scheme of fragment of installation.
General Idea The part of the tank which is in a shape conical was divided into three sections. Dividing idea is widely known and used (Dougherty and Cooper, 2003; Bhuvaneswari et al., 2008). Observations show there is no need to increase quantity of sections in studied tank. It is obvious the higher number of divided parts is the higher computational cost is. Consequently the higher accuracy is obtained (Dougherty and Cooper, 2003). However, received gain is to low so as to be taken into consideration. Therefore it was decided to use three sections. In each section the adaptive DMC controller was tuned. Additionally we have tuned the DMC controller above and below the cone. Unlike to the non-adaptive DMC controller which was only tuned for the lowest part of a cone. Depending on present liquid level appropriate matrix’s coefficients are changed - in such way as gain scheduling in PID control (Aström and Wittenmark, 1995).
3. DMC ALGORITHM Fig.2. Real laboratory installation. Predictive Control Mathematical Model According to the law of conservation of mass it can be written :
A(h)
dh = FIN − FOUT , dt
(1)
Where:
For predictive control (Tatjewski, 2002) in each instant k which is equal to the sampling time of a controller there is compute value of control u(k). This value depends on: -
Model of a process, Current and previous process values measurements, Previous controls, Set point in current instant k and future instants.
The u(k) vector is (Hc -1) length and has form as follows:
2 ⎧ ⎛d⎞ for h < H1 π⎜ ⎟ , ⎪ ⎝2⎠ ⎪ 2 ⎪⎪ ⎛ d D − d h − H1 ⎞ ⎟ , for H1 ≤ h < H 2 A(h) = ⎨π⎜⎜ + ⋅ ⎟ 2 H 2 − H1 ⎠ ⎪ ⎝2 2 ⎪ ⎛D⎞ for h ≥ H 2 π⎜ ⎟ , ⎪ ⎝2⎠ ⎩⎪
Table.1. Symbols description. Symbol
Value
Unit
Description
D d
12.3 1.48
cm cm
h
h(t)
cm
upper diameter lower diameter current liquid level
H1 H2 H3
65 85 120
cm cm cm
FIN
<0;40>
ml/s
FOUT
kV h
ml/s
geometrical dimensions inflow rate of the tank outflow rate of the tank
(2)
⎡ ⎤ u (k k ) ⎢ ⎥ ( ) u k + k 1 ⎢ ⎥ ⎥ u (k ) = ⎢ ⋅ ⎢ ⎥ ⋅ ⎢ ⎥ ⎢u (k + H − 1 k )⎥ C ⎣ ⎦
(3)
where Hc is the control horizon, which is one of the tuning parameters of the DMC algorithm. Notation (k + 1|k) means assigned control value in instant k for instant (k+1). In predictive algorithms control is assigned by minimization difference between set point ysp(k+p|k) and forecast in instant k process value y(k+p|k) on p steps to future. This minimization is done in discrete instants p on prediction horizon HP, where p = 1, 2, ... , HP. Objective function which is minimized is called criterion function. It consists of difference between set point trajectory and predicted trajectory of process value. Additionally it is a frequent rule to include the penalty for change of assigned control value. This penalty allows to influence on dynamic of control changes. The greater it is the more smooth control is
(less aggressive). Taking into consideration both previously mentioned components there can be formed a quadratic form of cost function for assigning control value in instant k. This form is mostly used.
[
HP
]
H C −1
J(k ) = ∑ ysp (k + p k ) − y(k + p k ) +λ ∑ [Δu (k + p k )] 2
p =1
2
(4)
p =0
Where:
ysp (k + p k ) - set point vector,
Δu(k + p k ) - evaluating increases of control, λ
- penalty for change of assigned control value,
y sp (k + p k ) - forecast values of process value depend on past
and future values of control signal, and past process values. DMC Algorithm Formulation
Next it is defined the system’s dynamic matrix G as: ⎡ g1 ⎢ g ⎢ 2 ⎢ M G = ⎢ ⎢g H C ⎢ M ⎢ ⎣⎢ g H P
i =1
∞
i = p +1
(5)
(6)
L O
g H P −1
L
(9)
y = G ⋅ Δu + y free
(10)
(
)
(
−1
Δu = G T G + λI G T ⋅ y sp − y free
)
(11)
(
Δu (k ) = K 1 ⋅ y sp − y free
)
(12)
Where K1 - it is a vector which contains first row of matrix K. For control is only selected first series of computed elements.
Afterwards it can be written:
y(k + p k ) = y free (k + p k ) + y forced (k + p k )
g H C −1 M
It can be written:
It is considered that disturbances are constant:
d (k + p k ) = d (k ) = y(k ) − y(k k − 1)
L O
Substituting (10) to cost function (4) and solving analytically we obtain:
p
i
g1 M
⎤ ⎥ 0 ⎥ ⎥ M ⎥ g1 ⎥ ⎥ M ⎥ g H P − H C + 1 ⎦⎥ 0
expression (9) can be written:
y(k + p k ) = ∑ g i ⋅ Δu (k + p − i ) +
∑ g ⋅ Δu(k + p − i ) + d(k + p k )
L
Where: dim G = HP x HC, HP - prediction horizon, HC - control horizon. This matrix was filled in with discrete elements of step response {g1, g2, g3, ... }.
According to (Camacho and Bordons, 1999) the DMC algorithm bases on step response of the process. Along the horizon predicted values can be written as:
+
0
(7) 4. EXPERIMENT RESULTS
Where:
y free (k + p k ) - trajectory of free response,
y forced (k + p k ) - trajectory of forced response,
p = 1, 2, ... , Hp then: HC
y(k + p k ) = ∑ g i ⋅ Δu (k + p − i ) + yfree (k + p | k ) i =1
(8)
The Adaptive DMC controller was tuned in five sections. As it was mentioned before. The part in shape of cone was divided into three parts. Fig.3. presents the way of partition. The Non-adaptive DMC controller was only tuned for the lowest part of a cone (section2 in Fig.3.). For each section step response was done. Obtained points were used for process tuning of DMC controller. The DMC controllers were tuned accordingly to tuning sequence of the DMC controller which was proposed by (Kłopot and Metzger, 2005; Kłopot et al., 2007).
Sequence is as follows: -
STEP1. Select the sample time T T = 0.1T1
-
STEP2. Compute the prediction horizon HP ⎛T ⎞ ⎛T ⎞ H P = round⎜ 1 ⎟ + round⎜ 0 ⎟ T ⎝ ⎠ ⎝T⎠
-
STEP3. Compute the control horizon HC HC = 2
-
STEP4. Compute the model horizon HD: ⎛T ⎞ ⎛ 2T ⎞ H D = round ⎜ 1 ⎟ + round ⎜ 0 ⎟ ⎝T⎠ ⎝ T ⎠
-
STEP5. Compute the move suppression coefficient λ: λ = HP ⋅ K2 ⋅ x
As opposed to the real-life installation. Both the nonlinearities of the plant and measurement noises can be distinguished, i.e. dead time and time variance of pumps and valves. Since the plant and controllers were simulated there can be done objective comparison between adaptive and nonadaptive DMC controller. As a comparison was used the time ratio, which is needed to achieve SP by PV. There was assumed if measured value PV is different from SP in some assumed range the time ratio will not be calculated. For each change of process set point the time ratio was calculated and presented in Table 2. In figures form Fig.4. to Fig.11. obtained process values for Non-Adaptive and Adaptive DMC are showed.
where: x - fine tuning parameter, T0 - process dead time, T1 - process time constant, K - steady state process gain. The dynamic of the process response can be modelled by means of the fine λ. By using this factor it is possible to have an effect on way of control signal changes. In this work parameter x, that is a part of λ formula, is equal to three. It is caused by requirements which were put on the process response. It was assumed that first overshoot of obtained process responses should to be less than 10% of SP change. In Fig.4, 6, 8, 10 mentioned interval was marked. The SP changes were made in both directions so as to take into account different behaviour of the process in each direction it is because of the process nonlinearity. In Fig.4. to Fig.7. changes of SP were situated in each section range. Whereas In Fig.8. to Fig.11. changes of SP were done between sections. Process responses of Non-Adaptive DMC was distinguished form Adaptive DMC by dotting the PV and the CV values. For better comparison PV and CV values were both plotted in separate charts - for Non-Adaptive and Adaptive DMC. This situation is presented in figures from Fig.4. to Fig.7. In this simulation natural limitations such as: signal control range and maximal volume of tank were considered by cutting signal value to allowed range. In (Tatjewski, 2002) there are showed other methods that are more sophisticated. The plant simulator and both controllers were programmed in LabView environment which provide real-time simulations. Consequently control system is free of all kind of noises and disturbances. Simulation also provides such additional properties as the repeatability of measurements and time invariance of the plant.
Fig.3. The division of the plant into sections.
Table.2. The time ratio values. The time ratio [s] Figure Number Fig.4. Fig.5. Fig.6. Fig.7. Fig.8. Fig.9. Fig.10. Fig.11.
Set Point Value
Adaptive DMC
NonAdaptive DMC
61
69.3
408
92
5026.8
4637.6
61→72, 72→77, 77→83, 83→90
616.1 1394 3211.2 4990.4
625.6 1557.2 3495.2 4868.8
95
60
94
58
93
56
92
54
91
H [cm]
H [cm]
62
52 50 48 46 44 0
100
200
300
400 t [s]
500
90
PV A DMC PV N-A DMC PVacc HI
89
PVacc LO
87
SP
86
600
PV A DMC PV N-A DMC PVacc HI
88
700
PVacc LO SP 9
Fig.4. Control quality for SP = 61[cm] in section 1.
9.2
9.4
9.6 t [s]
9.8
10
10.2 4
x 10
Fig.6. Control quality for SP = 92[cm] in section 5. 28
21 27.5 20.5
Qin [mL/s]
Qin [mL/s]
27 20
19.5
26.5 26 25.5
19
25
CV A DMC CV N-A DMC
18.5
0
100
200
300
400 t [s]
500
600
CV A DMC CV N-A DMC
24.5
700
9.2
Fig.5. Control signal for SP = 61[cm] in section 1.
9.4
9.6 t [s]
9.8
10
10.2 4
x 10
Fig.7. Control signal for SP = 92[cm] in section 5.
90
90 85
85 80
H [cm]
H [cm]
80 75
70
70
PV A DMC SP PVacc HI
65
75
PV N-A DMC SP PVacc HI
65
PVacc LO
PVacc LO
60
60 0
0.5
1
1.5 t [s]
2
2.5
3
2000
4000
6000
4
x 10
Fig.8. Control quality for SP = {72,77,83,90} [cm].
8000
10000 12000 14000 16000 18000 t [s]
Fig.10. Control quality for SP = {72,77,83,90} [cm].
25.5
28
25 27
24.5
26 Qin [mL/s]
Qin [mL/s]
24 23.5 23
25 24
22.5 23
22
22
21.5
CV N-A DMC
CV A DMC 21 0
0.5
1
1.5 t [s]
2
2.5
3
0 4
x 10
Fig.9. Control signal for SP = {72,77,83,90} [cm].
2000
4000
6000
8000 10000 12000 14000 16000 18000 t [s]
Fig.11. Control signal for SP = {72,77,83,90} [cm].
Table.3. Used nomenclature. Abbreviation
Description
SP
set point value
PV A DMC PV N-A DMC CV A DMC CV N-A DMC PVaccHI PVaccLO
process value for Adaptive DMC process value for Non-Adaptive DMC control value for Adaptive DMC control value for Non-Adaptive DMC upper and lower margin of permitted 10% overshoot
5. EXPERIMENT RESULTS
Apart from all advantages of predictive control the Non-Adaptive DMC controller can not deal with presented process with assumed quality. Taking into consideration a comparison between the time rations and control quality of studied controllers in relation to studied plant - it appears that the Non-Adaptive version of DMC is not satisfying - although the rations are nearly the same. It is brought out by strong and frequent changes of control signal value. Such a control is energy-consuming and it might turn out not possible to put into practice. Most of actuators do not support obtained character of signal control changes. For process control there was used an analytical version of the DMC algorithm where minimization of criterion function is done once in the stage of preparation. Consequently PLC controller implementation is possible. Both controllers were tuned accordingly to the tuning sequence presented in (Kłopot and Metzger, 2005; Kłopot et al., 2007). Proposed idea of dividing conical part of the tank into three sections combined with mentioned tuning sequence of the DMC controller allow to achieve quite attractive results. It has to be emphasized that controlled plant is strongly nonlinear by reason of the shape of the tank and turbulent outflow. Nevertheless obtained overshoots are less than 10%. Provided theoretical investigations and practical experiments on simulated plant proved correctness of proposed strategy.
ACKNOWLEDGEMENTS
This work was supported by the Polish Ministry of Scientific Research and Higher Education.
REFERENCES Aström, K. J., Hagglund, T. (1996). PID Control, The Control Handbook, edited by W. S. Levine. CRC Press, New York. Aström K. J., Wittenmark. (1995). Adaptive Control. Bhuvaneswari N.S., Uma G., Rangaswamy T.R. (2008). Adaptive and optimal control of a non-linear process using intelligent controllers, Elsevier 182-190 Camacho E.F. and Bordons C. (1999). Model Predictive Control, chapter 3, Springer, London. Cooper D. J. (2004). Practical Process Control Using Control Station, Control Station LLC, Storrs. Cutler C. R., Ramaker B.L. (1980). Dynamic matrix control a computer control algorithm. AIChE National Meeting, Houston, Texas, 1979. Proc. of Joint. Aut. Control Conf., San Francisco, California. Dougherty D., Cooper D. (2003). A practical multiple model adaptive strategy for single-loop MPC, Control Engineering Practice 11 141–159. Kłopot T., Metzger M. (2005). Simple Engineering Tuning of DMC Controller”, Proceedings of the 11th IEEE
International Conference on Methods and Models in Automation and Robotics, Międzyzdroje. Kłopot T., Łaszczyk P., Metzger M. (2007). Dynamic Matrix Control for Heating Process, 14th IFAC
International Conference on Methods and Models in Automation and Robotics, Międzyzdroje. Metzger M. (2000). Modeling, simulation and control of continuous processes, Edition of Jacek Skalmierski, Gliwice. Metzger M. (2007). Hydraulic pilot-plant with strong nonlinearities. Institute of Automatic Control Activity Report (2005-2006), Editions of Silesian University of Technology, Gliwice, pp. 1-212. Qin S.J. and Badgwell T.A. (2003). A survey of industrial model predictive control technology. Control Engineering Practice 11, 733–764 Sastry S., Bodson M. (1989). Adaptive Control Stability, Prentice-Hall Convergence and Robustness, Advanced Reference Series. Tatjewski P. (2002). Advanced control of industrial processes. Structures and Algorithms, Akademicka Oficyna Wydawnicza EXIT, Warszawa Vrancic D., Strmcnik S., Kocijan J. (2004) Improving disturbance rejection of PI controllers by means of the magnitude optimum method. ISA Trans; 43:73–84.
1. INTRODUCTION Control of flow and liquid level is a common and important task in process industries. There are many examples where such a control is used e.g.: petrochemical industries, paper making industries, water treatment industries etc. In each step of a process there can be found tanks which size and shape is dictated by process technology requirements e.g. for corrosive liquids could be used a shape of sphere where lateral area is the smallest in comparison to volume. It is important when is taken into consideration an expense of construction. Besides sphere shape can stand greater tension than other shapes. The other example which impose a great requirements for process control can be a conical tank. It is widely used in industries (Bhuvaneswari et al., 2008) e.g. food process industries, hydrometallurgical industries and especially in water treatment industries where it improves disposal of solids while mixing. This shape is a great challenge for process control. Conical tank will be considered in this paper. It was especially chosen because of its nonlinearities. This nonlinearities causing that plant parameters change more than two ranks. Obviously, the advanced PID controller could be used, but in spite of its common known advantages the quality of control could be not satisfied. Therefore, it was decided to choose Dynamic Matrix Control (DMC) algorithm (Cutler and Ramaker, 1980; Cooper, 2004) and show its comparison between adaptive and non-adaptive version. That kind of choice was done because in some special cases traditional algorithms do not deal with difficult processes and then raise a necessary of using advanced algorithm control. Among all advanced algorithms the predictive control is the most frequently used (Qin and Badgwell, 2003), where DMC algorithms enjoy a great popularity. Predictive control has many advantages nevertheless its number of practical implementation is still smaller than the number of practical applications of conventional controllers PID. Conventional controllers are still playing a main role in control applications (Aström and Hagglund, 1996; Vrancic et al., 2004). It is caused by more complicated construction of an advanced algorithm and also
by difficulties of controller tuning. We should take into account that each attempt which can emphasize economical gain is worth doing. That is why we decided to choose this algorithm. In this work we will base on “Tuning sequence of the DMC controller” which was proposed by (Kłopot and Metzger, 2005; Kłopot et al., 2007). Simulation studies and real-time DMC are discussed. 2. CONTROL PLANT DESCRIPTION
Controlled system is simulated (Metzger, 2000) in LabView environment which provides real-time simulation. The conical tank (Fig.1.) is a part of real laboratory installation (Metzger, 2007) (Fig.2.). It is assumed process value (PV) would be a level liquid and would be maintained by changing the input flow – control value (CV). Operating point is set by valve.
Fig.1. Scheme of fragment of installation.
General Idea The part of the tank which is in a shape conical was divided into three sections. Dividing idea is widely known and used (Dougherty and Cooper, 2003; Bhuvaneswari et al., 2008). Observations show there is no need to increase quantity of sections in studied tank. It is obvious the higher number of divided parts is the higher computational cost is. Consequently the higher accuracy is obtained (Dougherty and Cooper, 2003). However, received gain is to low so as to be taken into consideration. Therefore it was decided to use three sections. In each section the adaptive DMC controller was tuned. Additionally we have tuned the DMC controller above and below the cone. Unlike to the non-adaptive DMC controller which was only tuned for the lowest part of a cone. Depending on present liquid level appropriate matrix’s coefficients are changed - in such way as gain scheduling in PID control (Aström and Wittenmark, 1995).
3. DMC ALGORITHM Fig.2. Real laboratory installation. Predictive Control Mathematical Model According to the law of conservation of mass it can be written :
A(h)
dh = FIN − FOUT , dt
(1)
Where:
For predictive control (Tatjewski, 2002) in each instant k which is equal to the sampling time of a controller there is compute value of control u(k). This value depends on: -
Model of a process, Current and previous process values measurements, Previous controls, Set point in current instant k and future instants.
The u(k) vector is (Hc -1) length and has form as follows:
2 ⎧ ⎛d⎞ for h < H1 π⎜ ⎟ , ⎪ ⎝2⎠ ⎪ 2 ⎪⎪ ⎛ d D − d h − H1 ⎞ ⎟ , for H1 ≤ h < H 2 A(h) = ⎨π⎜⎜ + ⋅ ⎟ 2 H 2 − H1 ⎠ ⎪ ⎝2 2 ⎪ ⎛D⎞ for h ≥ H 2 π⎜ ⎟ , ⎪ ⎝2⎠ ⎩⎪
Table.1. Symbols description. Symbol
Value
Unit
Description
D d
12.3 1.48
cm cm
h
h(t)
cm
upper diameter lower diameter current liquid level
H1 H2 H3
65 85 120
cm cm cm
FIN
<0;40>
ml/s
FOUT
kV h
ml/s
geometrical dimensions inflow rate of the tank outflow rate of the tank
(2)
⎡ ⎤ u (k k ) ⎢ ⎥ ( ) u k + k 1 ⎢ ⎥ ⎥ u (k ) = ⎢ ⋅ ⎢ ⎥ ⋅ ⎢ ⎥ ⎢u (k + H − 1 k )⎥ C ⎣ ⎦
(3)
where Hc is the control horizon, which is one of the tuning parameters of the DMC algorithm. Notation (k + 1|k) means assigned control value in instant k for instant (k+1). In predictive algorithms control is assigned by minimization difference between set point ysp(k+p|k) and forecast in instant k process value y(k+p|k) on p steps to future. This minimization is done in discrete instants p on prediction horizon HP, where p = 1, 2, ... , HP. Objective function which is minimized is called criterion function. It consists of difference between set point trajectory and predicted trajectory of process value. Additionally it is a frequent rule to include the penalty for change of assigned control value. This penalty allows to influence on dynamic of control changes. The greater it is the more smooth control is
(less aggressive). Taking into consideration both previously mentioned components there can be formed a quadratic form of cost function for assigning control value in instant k. This form is mostly used.
[
HP
]
H C −1
J(k ) = ∑ ysp (k + p k ) − y(k + p k ) +λ ∑ [Δu (k + p k )] 2
p =1
2
(4)
p =0
Where:
ysp (k + p k ) - set point vector,
Δu(k + p k ) - evaluating increases of control, λ
- penalty for change of assigned control value,
y sp (k + p k ) - forecast values of process value depend on past
and future values of control signal, and past process values. DMC Algorithm Formulation
Next it is defined the system’s dynamic matrix G as: ⎡ g1 ⎢ g ⎢ 2 ⎢ M G = ⎢ ⎢g H C ⎢ M ⎢ ⎣⎢ g H P
i =1
∞
i = p +1
(5)
(6)
L O
g H P −1
L
(9)
y = G ⋅ Δu + y free
(10)
(
)
(
−1
Δu = G T G + λI G T ⋅ y sp − y free
)
(11)
(
Δu (k ) = K 1 ⋅ y sp − y free
)
(12)
Where K1 - it is a vector which contains first row of matrix K. For control is only selected first series of computed elements.
Afterwards it can be written:
y(k + p k ) = y free (k + p k ) + y forced (k + p k )
g H C −1 M
It can be written:
It is considered that disturbances are constant:
d (k + p k ) = d (k ) = y(k ) − y(k k − 1)
L O
Substituting (10) to cost function (4) and solving analytically we obtain:
p
i
g1 M
⎤ ⎥ 0 ⎥ ⎥ M ⎥ g1 ⎥ ⎥ M ⎥ g H P − H C + 1 ⎦⎥ 0
expression (9) can be written:
y(k + p k ) = ∑ g i ⋅ Δu (k + p − i ) +
∑ g ⋅ Δu(k + p − i ) + d(k + p k )
L
Where: dim G = HP x HC, HP - prediction horizon, HC - control horizon. This matrix was filled in with discrete elements of step response {g1, g2, g3, ... }.
According to (Camacho and Bordons, 1999) the DMC algorithm bases on step response of the process. Along the horizon predicted values can be written as:
+
0
(7) 4. EXPERIMENT RESULTS
Where:
y free (k + p k ) - trajectory of free response,
y forced (k + p k ) - trajectory of forced response,
p = 1, 2, ... , Hp then: HC
y(k + p k ) = ∑ g i ⋅ Δu (k + p − i ) + yfree (k + p | k ) i =1
(8)
The Adaptive DMC controller was tuned in five sections. As it was mentioned before. The part in shape of cone was divided into three parts. Fig.3. presents the way of partition. The Non-adaptive DMC controller was only tuned for the lowest part of a cone (section2 in Fig.3.). For each section step response was done. Obtained points were used for process tuning of DMC controller. The DMC controllers were tuned accordingly to tuning sequence of the DMC controller which was proposed by (Kłopot and Metzger, 2005; Kłopot et al., 2007).
Sequence is as follows: -
STEP1. Select the sample time T T = 0.1T1
-
STEP2. Compute the prediction horizon HP ⎛T ⎞ ⎛T ⎞ H P = round⎜ 1 ⎟ + round⎜ 0 ⎟ T ⎝ ⎠ ⎝T⎠
-
STEP3. Compute the control horizon HC HC = 2
-
STEP4. Compute the model horizon HD: ⎛T ⎞ ⎛ 2T ⎞ H D = round ⎜ 1 ⎟ + round ⎜ 0 ⎟ ⎝T⎠ ⎝ T ⎠
-
STEP5. Compute the move suppression coefficient λ: λ = HP ⋅ K2 ⋅ x
As opposed to the real-life installation. Both the nonlinearities of the plant and measurement noises can be distinguished, i.e. dead time and time variance of pumps and valves. Since the plant and controllers were simulated there can be done objective comparison between adaptive and nonadaptive DMC controller. As a comparison was used the time ratio, which is needed to achieve SP by PV. There was assumed if measured value PV is different from SP in some assumed range the time ratio will not be calculated. For each change of process set point the time ratio was calculated and presented in Table 2. In figures form Fig.4. to Fig.11. obtained process values for Non-Adaptive and Adaptive DMC are showed.
where: x - fine tuning parameter, T0 - process dead time, T1 - process time constant, K - steady state process gain. The dynamic of the process response can be modelled by means of the fine λ. By using this factor it is possible to have an effect on way of control signal changes. In this work parameter x, that is a part of λ formula, is equal to three. It is caused by requirements which were put on the process response. It was assumed that first overshoot of obtained process responses should to be less than 10% of SP change. In Fig.4, 6, 8, 10 mentioned interval was marked. The SP changes were made in both directions so as to take into account different behaviour of the process in each direction it is because of the process nonlinearity. In Fig.4. to Fig.7. changes of SP were situated in each section range. Whereas In Fig.8. to Fig.11. changes of SP were done between sections. Process responses of Non-Adaptive DMC was distinguished form Adaptive DMC by dotting the PV and the CV values. For better comparison PV and CV values were both plotted in separate charts - for Non-Adaptive and Adaptive DMC. This situation is presented in figures from Fig.4. to Fig.7. In this simulation natural limitations such as: signal control range and maximal volume of tank were considered by cutting signal value to allowed range. In (Tatjewski, 2002) there are showed other methods that are more sophisticated. The plant simulator and both controllers were programmed in LabView environment which provide real-time simulations. Consequently control system is free of all kind of noises and disturbances. Simulation also provides such additional properties as the repeatability of measurements and time invariance of the plant.
Fig.3. The division of the plant into sections.
Table.2. The time ratio values. The time ratio [s] Figure Number Fig.4. Fig.5. Fig.6. Fig.7. Fig.8. Fig.9. Fig.10. Fig.11.
Set Point Value
Adaptive DMC
NonAdaptive DMC
61
69.3
408
92
5026.8
4637.6
61→72, 72→77, 77→83, 83→90
616.1 1394 3211.2 4990.4
625.6 1557.2 3495.2 4868.8
95
60
94
58
93
56
92
54
91
H [cm]
H [cm]
62
52 50 48 46 44 0
100
200
300
400 t [s]
500
90
PV A DMC PV N-A DMC PVacc HI
89
PVacc LO
87
SP
86
600
PV A DMC PV N-A DMC PVacc HI
88
700
PVacc LO SP 9
Fig.4. Control quality for SP = 61[cm] in section 1.
9.2
9.4
9.6 t [s]
9.8
10
10.2 4
x 10
Fig.6. Control quality for SP = 92[cm] in section 5. 28
21 27.5 20.5
Qin [mL/s]
Qin [mL/s]
27 20
19.5
26.5 26 25.5
19
25
CV A DMC CV N-A DMC
18.5
0
100
200
300
400 t [s]
500
600
CV A DMC CV N-A DMC
24.5
700
9.2
Fig.5. Control signal for SP = 61[cm] in section 1.
9.4
9.6 t [s]
9.8
10
10.2 4
x 10
Fig.7. Control signal for SP = 92[cm] in section 5.
90
90 85
85 80
H [cm]
H [cm]
80 75
70
70
PV A DMC SP PVacc HI
65
75
PV N-A DMC SP PVacc HI
65
PVacc LO
PVacc LO
60
60 0
0.5
1
1.5 t [s]
2
2.5
3
2000
4000
6000
4
x 10
Fig.8. Control quality for SP = {72,77,83,90} [cm].
8000
10000 12000 14000 16000 18000 t [s]
Fig.10. Control quality for SP = {72,77,83,90} [cm].
25.5
28
25 27
24.5
26 Qin [mL/s]
Qin [mL/s]
24 23.5 23
25 24
22.5 23
22
22
21.5
CV N-A DMC
CV A DMC 21 0
0.5
1
1.5 t [s]
2
2.5
3
0 4
x 10
Fig.9. Control signal for SP = {72,77,83,90} [cm].
2000
4000
6000
8000 10000 12000 14000 16000 18000 t [s]
Fig.11. Control signal for SP = {72,77,83,90} [cm].
Table.3. Used nomenclature. Abbreviation
Description
SP
set point value
PV A DMC PV N-A DMC CV A DMC CV N-A DMC PVaccHI PVaccLO
process value for Adaptive DMC process value for Non-Adaptive DMC control value for Adaptive DMC control value for Non-Adaptive DMC upper and lower margin of permitted 10% overshoot
5. EXPERIMENT RESULTS
Apart from all advantages of predictive control the Non-Adaptive DMC controller can not deal with presented process with assumed quality. Taking into consideration a comparison between the time rations and control quality of studied controllers in relation to studied plant - it appears that the Non-Adaptive version of DMC is not satisfying - although the rations are nearly the same. It is brought out by strong and frequent changes of control signal value. Such a control is energy-consuming and it might turn out not possible to put into practice. Most of actuators do not support obtained character of signal control changes. For process control there was used an analytical version of the DMC algorithm where minimization of criterion function is done once in the stage of preparation. Consequently PLC controller implementation is possible. Both controllers were tuned accordingly to the tuning sequence presented in (Kłopot and Metzger, 2005; Kłopot et al., 2007). Proposed idea of dividing conical part of the tank into three sections combined with mentioned tuning sequence of the DMC controller allow to achieve quite attractive results. It has to be emphasized that controlled plant is strongly nonlinear by reason of the shape of the tank and turbulent outflow. Nevertheless obtained overshoots are less than 10%. Provided theoretical investigations and practical experiments on simulated plant proved correctness of proposed strategy.
ACKNOWLEDGEMENTS
This work was supported by the Polish Ministry of Scientific Research and Higher Education.
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