2015 23rd Iranian Conference on Electrical Engineering (ICEE)
Tuning of Generalized Predictive Controllers for First Order plus Dead Time Models Based on ANOVA Zahed Ebrahimi
Peyman Bagheri
Center of Excellence in Industrial Control, Department of Electrical Engineering, K. N. Toosi University of Technology, Tehran, Iran.
[email protected]
where, yÖ is the future output value, y d is the desired
Abstract: Successful implementation of predictive controller requires an appropriate tuning of its parameters. Closed form tuning equations are practically rewarding as they can be easily implemented with relatively low computational costs. In this paper, a tuning strategy for the generalized predictive control of single input-single output and multi inputmulti output plants is presented. First order plus dead time model of the plant is considered and analysis of variance and nonlinear fitting is employed to derive tuning equations. Finally, simulation results are used to verify the efficiency of the proposed tuning strategy.
output, 'u is the control effort, N 1 and N 2 are the lower and upper values of the prediction horizon respectively, N u is the control horizon and O is the moving suppression factor. Optimizing cost function (1) due to prediction values of output leads to optimal control effort as
'u
j N1
N u 1
O ¦ ǻu t j
1
G Te
(2)
In [6], a study on First Order plus Dead Time (FOPDT) models is performed to obtain the most effective MPC tuning parameters on system performance and it is shown that the moving suppression coefficient ( O ) is the most effective parameter in the closed loop performance. Then, an analytical equation for O is presented to avoid singularity in control effort calculations. This analytical method is extended to multivariable plants in [7] and integrated processes in [8] with the same procedure. A tuning strategy for Second Order plus Dead Time (SOPDT) models is presented in [9]. State-space analytical tuning methodology is performed in [10] for FOPDT models. In [10], the issues of closed-loop stability and possible achievable performance are addressed. Also, it is shown that for the FOPDT models control horizon of two provides the maximum achievable performance. The proposed method in [10] is extended for unstable plants with fractional dead time and multivariable plants in [11] and [12] respectively. Some efforts have been done in on-line tuning like gradient
(1)
2
j 0
c 978-1-4799-1972-7/15/$31.00 2015 IEEE
system performance [6]. In this paper, the closed form equations will be presented to tune O . Note that, MPC tuning problem is not straight forward due to complex and nonlinear relations between the tuning parameters and closed loop characteristics.
2
d
G OI
N u . It is shown that O is more influent on the closed loop
MPC parameters tuning is a challenging issue in achieving high-quality performance of the controller. The available tuning methods try to find a simple tuning methodology for practical implementations. A typical form of objective function in GPC is N2
T
and G is a dynamic matrix that is obtained from a Diophantine equation. Note that, according to (1) and (2), tunable parameters of GPC algorithm are O , N 1 , N 2 and
I. Introduction Model Predictive Control (MPC) is a well-known advanced process control strategy, which has attracted interest and attention over the recent decades [1, 2]. In MPC, control sequences are calculated by optimizing an objective function due to output prediction values [3]. Future output predictions are calculated using a proper model. Transfer functions are commonly used models in the Generalized Predictive Control (GPC) algorithm [4]. Other common models are step response model which is used in Dynamic Matrix Control (DMC) and state-space models. In the MPC family, the most popular method is DMC which is widely used in many chemical processes. This popularity is due to the fact that DMC uses step response information in calculating predictions which is easily obtained for stable industrial process [5].
¦ yÖ t j | t y t j
G
where 'u is future values of the control effort vector over the control horizon N u , e is the vector of predicted errors
Keywords: Model predictive control; generalized predictive control; controller tuning; analysis of variances; first order plus dead time models
J
Ali Khaki-Sedigh
Control Engineering Department, Center of Excellence in Industrial Faculty of Electrical and Computer Control, Department of Electrical Engineering, University of Engineering, K. N. Toosi University Tabriz, Tabriz, Iran. of Technology, Tehran, Iran.
[email protected] [email protected]
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2015 23rd Iranian Conference on Electrical Engineering (ICEE) decrement, fuzzy logic and on-line optimization algorithms. For example, in [13] using gradient decrement methods an algorithm is presented for MPC tuning problem. In [14], by defining some membership functions a tuning method is proposed for O . Another suggested on-line tuning method for O is based on Particle Swarm Optimization (PSO) algorithm [15]. Some researchers tried to improve the robustness of MPC by tuning [16]. Robust performance number is used in [17] to achieve a tuning strategy is for non minimum phase multivariable plants. Also, in [18] by calculating the open loop frequency response and desired close loop frequency response, MPC tuning problem is solved for MIMO plants. Another approach to MPC tuning problem is using Analysis of Variance (ANOVA). First in [19] an equation for O is obtained using ANOVA. The errors in obtaining tuning equation in [19] are corrected in [5] and the results are also improved. Also, [20] used this tuning procedure for SOPDT plants. In this paper, the tuning procedure based on ANOVA and nonlinear regression is used for GPC algorithm for both SISO and MIMO plants and closed form tuning equations are obtained. Note that, these equations are useful in closed loop studies.
Now, the goal is finding the optimal value of f . To realize this goal, a cost function is defined as t
Jc
d
t
dt ī ³ ǻu t dt 2
(5)
0
o
* is chosen as Table I. Table I. Setup of parameters for ANOVA in SISO model ANOVA parameters
Level Low
W
2 2
td
*
0.1
Level Low Medium 5 5 0.3
Level Medium
Level High Medium 40 20
10 10
0.5
1
2
Level High 80 25 3
4
Then, for each model and * according to the table I, optimal value of f , i.e. f opt is obtained. After finding f opt , ANOVA [21] is performed on f opt as a response vector and model parameters as variables. Results of ANOVA show which one of the model parameters or a combination of the model parameters is more influent on the optimal parameter f opt and also the level of influence is determined by ANOVA. In table II the results of ANOVA are presented. TableII. ANOVA result in GPC Source
Sum Squares 0.064
Degree of Freedom 16
Mean Squares 0.004
F value
Pvalue
6.5748e+11
0
355.22
6
59.2033
9.721e+15
0
0.053
96
0.0006
9.0435e+10
0
Error
0
56
0
Total
409.459
174
II. Tuning procedure for SISO plant Consider the following FOPDT model
td
W
*
ke t d s (3) G s W s 1 where k is the steady-state gain, t d is the dead time and
td
W
W
is the time constant. This model is used to achieve closed form tuning equations. In the GPC tuning problem, the prediction horizon and control horizon must be large enough to have robustness [16] and the effect of system output and controller output have to be considered during process to calculate the proper control signal. In the GPC algorithm, N 2 is chosen such that it covers the settling time of open
u*
In this table, there are F-values and P-values associated with each parameter and combination of parameters. These values reveal the effect and also the effectiveness level of these parameters on optimal parameters. Typically there is a cut-off value of 0.05 for P index. That is, any of these sources having a value below the cut-off is considered to be significant. Also, a source with small P-value and larger Fvalue has larger influence on optimal tuned parameter. The remaining terms are omitted. Many nonlinear combinations were tested. Finally, the combination which is depicted in Table II was more effective. t d and * combination for W tuning f was used in [5] in the DMC algorithm. Here f is tuned for the GPC algorithm. This tuning equation is as
loop system. N 1 is equal to d 1 , where d is discrete samples of dead time. Finally, the control horizon N u is chosen equal as N 2 d . According to (1), the last parameter which is to be tuned is O . Note that, this parameter can be normalized due to the steady-state gain ( k ) [6]. The normalization is as following f uk 2
2
where * is a parameter which shows the importance of the control effort. Now a class of model parameters W , t d and
This paper is organized as following. In section 2 the tuning strategy is proposed for SISO plants. Also, the effects of reference trajectory filtering are proposed in obtaining tuning equations. In section 3, ANOVA tuning procedure is developed for two-input two-output systems. In these sections, the effectiveness of tuning equations is tested via some examples. The last section concludes the paper.
O
³ yÖ t y t
(4)
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2015 23rd Iranian Conference on Electrical Engineering (ICEE)
fÖopt
E3
§ td · E 2 ¸ ī E4 W © ¹ 2.9906 E3 0.0246 E 4
1.5
E1 ¨
Reference Schwarz [22] Proposed Bagheri [5]
(6)
E1 1.0905 E 2 0.9848 In Fig 1, the accuracy of proposed tuning equation is shown. As it is seen the equation for fÖopt has good enough accuracy. y
1
Now the performance of tuning equations will be compared with an example from [22] where a tuning equation is given for GPC and also the comparison is performed with [5] and [22]. Consider the following transfer function
1
0.5
0
0 13.5
e s (7) G s 0.5s 1 where T s , N 1 , N 2 , N u and O were taken 0.5, 3, 9, 3 and 1.1, respectively according to the method of [22]. For the proposed method in this paper, the parameters are 0.075, 15, 41, 27 and 2.245 by assuming * 2 . In [5] the model, prediction and control horizons, T s and O are respectively equal to 121, 121, 4, 0.05 and 1.843, respectively. In Fig 2, the closed loop responses are presented. The ISE criteria in [22], [5] and the proposed method are respectively equal to 7.12, 4.032 and 4.26. The proposed method and [5] almost have the same performance but note that [5] leads to greater horizons. 5
0.5
-0.5 10
12
14
14
16
14.5
18
20 Time(s)
15
22
15.5
16
24
26
28
30
2 Schwarz [22] Proposed Bagheri [5]
1.5
u
1
0.5
1.5
0
0.5
1
Estimated Real
0 11.5
12.5
13
18
3
20 Time(s)
2.5
Fig 2. Comparison of system performance
4.5
12
13.5
14
-0.5
4
f
Opt
3.5
-1 10
2
12
14
16
22
24
26
28
30
Again using analysis of variance and nonlinear regression, the tuning formula is achieved as
1.5 1
0.5 0
10
20
30 Model
40
50
60
E1 E5 E9
Fig 1. Accuracy of GPC tuning equation
Sharp changes in reference trajectory could cause overshoot in both control signal and output while by proper filtering of reference trajectory the overshoot would be avoided. This method changes reference edges to a smooth corner. We have
yÖ d t
1 D y d t 1 D yÖ d t 1
(8)
where D could be chosen between zero and one. This method affects f opt value on each model. Here by defining various value for W , t d , D and * as table III some experiences were done to realize the effect of D on f opt .
930
td
E11
§t · E1ī D e E5 ī e E9 ī ¨ d ¸ (9) ©W ¹ 0.317 E 2 1.26 E3 0.422 E 4 0.113
fÖopt
E2
E3
E4
W
E6
E7D E8
E10
E 6 0.92 E 7 208.064 E8 0.029 E10 1.551 E11 0.202
0.852
5.071
2015 23rd Iranian Conference on Electrical Engineering (ICEE) III. Tuning procedure for MIMO plants In the MIMO tuning procedure, it is attempted to find appropriate simple equations. For multivariable systems, objective function is
1.5 Reference y Filtered Reference Filtered Reference y 1
N2
¦ yÖ t j | t y t j
J
d
2 Q
y
j N1
N u 1
0.5
(10)
¦ ǻu t j R
1
2
j 0
In dealing with 2-input 2-output systems, Q and R can be chosen as diag I, mI and diag O1I, O2I , respectively.
0.5
0 0 13.5
-0.5 10
12
14
14
16
14.5
18
20 Time (s)
15
22
where, I is an identity matrix with suitable dimension. By optimizing (10) due to predictions, optimal control effort sequence is obtained as follows
15.5
24
26
28
30
'u
T
QG RI
1
G T Qe
(11)
The FOPDT model for 2-input 2-output plant is as
1.5 u Filtered Reference u
G s
t d ij s
1, 2 (12)
i,j
W ij s 1
The following criteria is used to be optimized for achieving optimal tuning parameters
1.5
0.5
k ij e
gij s
ª¬ g ij (s ) º¼ ,
1
u
G
1
0.5 0 12.5
13
13.5
³ yÖ t y t
Jc
d
2 M
dt ³ ǻu t ī dt 2
(13)
Here ī is diag ī1 ī 2 and M is diag 1, m . First it is
14
assumed that the system is fixed and only *1 , * 2 and m are changed as table IV . The system is considered as
0
-0.5 10
12
14
16
18
20 Time (s)
22
24
26
28
1 ª2 2º « » s 1 ¬2 3¼
G s
30
Fig 3. Comparison of system performance in the filtered reference mode
(14)
Table IV. Setup parameters for ANOVA in the first step Table III. Setup of parameters for ANOVA in filtering mode ANOVA parameters
Level Low
W
2 2
td
D
*
Level Medium 10 10
0.1 2
3
0.3 4
ANOVA Parameters
Level High
Level Low
m
80 25
Level Medium
Level High
1.2
2
0.5
*1
0.8
1.5
2
2.5
*2
1.2
2.2
3
3.5
0.6 5
6
Now using ANOVA the suitable equations are estimated in the form of
To check the efficiency of this tuning equation, transfer function (7) is considered. We choose, * 2 and D 0.7 thus f opt is equal to 0.6192. The ISE criteria, without
OÖ1opt
OÖ2opt
filtering and with filtering is respectively equal to 4.26 and 3.49 and in the filtered mode the output has no overshoot and its response is more smooth. The results are shown in Fig 3.
E ī E ī
E2
1 1
1
E5
e E3ī2 E 4e ī1 ī2 E6 m E7
E2 2
E5
e E3ī1 E 4e ī2 ī1E6 m E7
(15) (16) 2
2
In the second step, in addition to *1 , * 2 and m , steadystate gain values ( k ij ) is changed as Table V. The structure of equations is as follows
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2015 23rd Iranian Conference on Electrical Engineering (ICEE)
OÖ1opt
E1e E k k 2 E ī1E e E ī 3
2 1
(17)
E 6 k 1E7 e E8 k 2 e E9 ī1 ī 2 E10 m E11
OÖ2opt
E1e E k k 2 E ī 2 E e E ī 3
2 1
,
k 11k 22
E9 0.062 E13 0.434 E17 0.577 W 1 W 11 uW 21
5 1
4
(18)
E 6 k 1E7 e E8 k 2 e E9 ī 2 ī1E10 m E11 k1
E1 1.568 E 2 0.417 E5 0.086 E 6 0.084
5 2
4
k2
k 12 k 21
Level Low 0.5
k ij
W ij
7
0.418 0.135
0.909 0.506
Level Low 2
Level Medium 3
Here, with an example the efficiency of proposed tuning equations is examined. Consider
VI the effect of this parameter on tuning procedure would be analyzed.
G s
Table VI. Setup parameter for ANOVA in the third step Level Low 0.4
0.165
E4 E8
E10 5.6 u10 E11 0.065 E12 E14 0.093 E15 0.28 E16 E18 2.215 E19 0.286 , W 2 W 12 uW 22
ANOVA Parameters
Level Medium 1.2
In the third step by changing t d value according to Table ij
ANOVA Parameters t d ij
0.673
Table VII. Setup parameter for ANOVA in the fourth step
Table V. Setup parameter for ANOVA in the second step ANOVA Parameters
E3 E7
Level Medium 1.2
ª 1.5e 2 s « « 3s 1 « 0.6e 3s «¬ 2s 1
0.6e 3s 2s 1 1.5e 2 s 3s 1
º » » » »¼
(23)
In this example, m , *1 and * 2 are chosen as 0.1, 1 and Again ANOVA is performed on data and various nonlinear combinations are tested and the final result is
OÖ1opt
E1e E k k 2 E ī1E e E ī e 3
2 1
E9
E10 k 2
E9
E10 k 2
E8 k 1 e
5 2
4
E6t d1 E7t d 2
E11ī1
E12
E13
E14t d1 E15t d 2
E11ī 2
E12
E13
E14t d1 E15t d 2
ī2 m e E t E t E1e E 2 k1 k 2 E3 ī 2 E 4 e E5 ī1e 6 d1 7 d 2
OÖ2opt
E8 k 1 e t d1
e
e
,
t d11t d12
0.7, respectively. The value of T s , N 1 , N 2 , N u and Q are respectively equal to 1, 3, 15, 13 and diag 1.69I, 0.83I . The performance of the proposed
ī1 m e
(19)
method is compared with the well-known tuning method in [7]. In which, the model, prediction and control horizons, T s and Q are respectively equal to 18, 18, 6, 1 and
(20)
diag 0.41I, 0.2I . The closed loop response of proposed method and the method of [7] is shown in Fig 4 and Fig 5.
t d 21t d 22
td2
The ISE criteria in the proposed method is equal to 11.33 and in [7] it is equal to 14.97. In the proposed method, the value of horizons is smaller, the system response is faster and the control signal is smoother and has less abrupt changes in its value than [7].
In the final step, by changing W ij value as Table VII the last experiment for tuning procedure was done. The ANOVA analysis is performed on various nonlinear combinations of these collected data and its result is suitable equations. These equations are
OÖ1opt
E6t d1 E7t d 2 E8W1 E9 e W2
E1e E k k 2 E ī1E e E ī e 2 1
3
5 2
4
E10 k 1E11e E12 k 2 e E13ī1 ī 2 E14 m E15 e
E16t d1 E17t d 2
E1 130.056 E 2 0.484 E3 0.217 E5 0.048 E 6 0.071 E 7 0.308 E9 1.893 E10 2044.7 E11 3.934 E13 0.437 E14 0.064 E15 0.326 E17 0.712 E18 7.543 E19 0.698 OÖ2opt
E1e E k k 2 E ī 2 E e E ī e 2 1
E11
E10 k 1 e
3
E12 k 2
4
e
5 1
E13 ī 2
W 1E e E W E 4 0.431 E8 0.157 E12 3.796 E16 0.085 18
19 2
E18
E19W 2
E6 t d1 E7 t d 2 E8W1 E9 e W2
E14
E15
ī1 m e
IV. Conclusion A new tuning strategy based on ANOVA and nonlinear regression is proposed for GPC for both SISO and MIMO plants. In the SISO case, by using filtered reference and including its parameter in the tuning equation, system performance is improved. The effectiveness of both SISO and multivariable tuning methods are shown be simulation results.
(21)
E16 t d1 E17 t d 2
(22)
W1 e
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2015 23rd Iranian Conference on Electrical Engineering (ICEE) [7] 1
1.5
[8]
0.5
y
1
1 0 220
0.5
230
240
250
260
[9]
0 -0.5
0
50
100
150
200
250
300
350
[10]
Reference Cooper [7] Proposed method
1.5
[11]
1
y
2
1
0.5
[12]
0.5
0 -0.5
0 45
0
50
100
50
55
60
150 200 Time (s)
65
70
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75
300
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[14]
FLJ6\VWHP¶VRXWSXWDFFRUGLQJWRMIMO tuning equations
[15]
1 Cooper [7] Proposed method
[16]
u
1
0.5
0
[17] -0.5
0
50
100
150
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250
300
350
[18] 1
[19]
u
2
0.5
0
-0.5
[20] 0
50
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250
300
350
)LJ6\VWHP¶VFRQWUROHIIRUWDFFRUGLQJWR0,02WXQLQJHTXDWLRQV
[21] [22]
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