Proceedings of the 10th Mediterranean Conference on Control and Automation - MED2002 Lisbon, Portugal, July 9-12, 2002.
IDENTIFICATION OF CONTINUOUS PROCESSES PARAMETERS USING GENETIC ALGORITHMS J.M. Herrero∗ , X. Blasco, M. Martínez, J. Sanchis† ∗
Predictive Control and Heuristic Optimization Group Department of Systems Engineering and Control Politechnic University of Valencia Camino de Vera 14, P.O. Box 22012 E-46071 Valencia, Spain fax: +34-963879579 e-mail:
[email protected], http://ctl-predictivo.upv.es †
MED2002 Conference e-mail:
[email protected] http://www.isr.ist.utl.pt/med2002/ Keywords: Genetic Algorithms, Optimization, Pro- in a state space representation. In these models, parameters have a meaning (it means prior informacess Identification, Non-linear Models. tion), which can be useful to validate the model. Behavioural models try to approximate process evoluAbstract tion without prior information, for instance, with a This work presents a technique for identifying the polynomial, a neural network, a fuzzy set, etc. Separameters of a continuous process using Genetic lection between these groups is not simple. This paAlgorithms. The flexibility of this technique al- per is focused on exploiting prior knowledge, and so lows the parameters identification of high order lin- the model will be obtained from first principles. ear models, and non-linear models. One of the adAnother aspect to take into account is the parameter vantages is that any type of input signal can be used. estimation. There is a well established set of idenThis feature is very useful when an industrial process tification techniques for linear discrete models and cannot be stopped for specific identification opera- for continuous low order models [7] [8], but identitions, and so regular process operations can be used fication is not so clear with non-linear or with confor identification. This paper shows the application tinuous high order models. In these cases, a paramof this technique to a thermal process and a conveyor eter identification problem becomes a complicated system. optimization problem. For these situations, a global optimization technique is necessary, and Genetic Al1 Introduction gorithms offer a very good approach for off-line optimization. One of the first steps in many technological areas consists of building a mathematical model. For instance, in process control, modelling crucially influ- 2 Genetic Algorithms ences on quality control. So building a linear, or Genetic Algorithms (GA) [3, 5] are optimization non-linear model, is a common, and often difficult techniques based on the simulation of the phenomproblem. ena which take place in the evolution of species. It is possible to divide models into two groups [8]: These techniques are based on applying natural sephenomenological and behavioural models. The for- lection laws onto a population to achieve individuals mer are obtained from the first principles and result which are better adjusted to their environment.
A population is nothing more than a set of points in the search space. Each individual of the population represents a point in that space by means of his chromosomes. The adaptation degree of the individual is given by the objective function (function to optimize).
• u(t): vector of model inputs (size m). • yˆ(t): vector of model outputs (size l). • x(t): vector of state variables (size n).
The identification objective consists of finding a set The evolution mechanism of individuals is achieved of parameters (ζ) ˆ which minimizes differences beby genetic operators. The usual operators are: tween the real response vector of the process (y(t)) and the model output vector (ˆ y (t)). Mathematically, • Selection: Its main goal consists of selecting it means a minimization problem of a function of the chromosomes to integrate the next popula- y(t) − yˆ(t) on a time interval (cost function): tion (these would depend on the cost function for each individual). J(ζ) = f (y(t) − yˆ(t)) (1) • Crossover: New individuals are generated and integrated by combining the chromosomes of Real response is obtained from experimentation using this methodology. There is no condition on sigtwo individuals. nal inputs, so the input signal are not forced to be a • Mutation: Randomly varying of some parts of white noise, step, etc, because prior information is the chromosome of an individual in the popula- included when the structure is selected. This property means that the information obtained from the tion generates new individuals. regular process operation can be used for identification. To improve model quality, if the process is The different possible variations of Genetic Algorithms can be distinguished by the chromosome cod- non-linear, information from the most characteristic process operation must be used. ification and the genetic operators used. GAs have demonstrated very good performance as Model outputs are obtained by the simulated appliglobal optimizers in many types of applications [9, cation to the model of the same inputs as used in the experiment. Since the model is continuous, a numer2, 1]. ical integration method is necessary for simulation (for instance, Runge-Kutta).
3 Parameters identification with GA
The most intuitive cost function for a minimization The proposed technique assumes a first principle in a time interval [0, te] (N samples) is: model structure where all, or some, of the parameters are unknown. In this case, the general formuN X l X lation of the model is a set of differential equations J(ζ) = kij |yi (j) − yˆi (j)| (2) which can be converted into a state space represenj=1 i=1 tation: Where: x(t) ˙ = f (x(t), u(t), ζ) yˆ(t) = g(x(y), u(t), ζ) Where: • f (·) , g(·): model functional structure. • ζ: vector of parameters to identify.
• N : number of samples. • kij : weighting factors. • yi (j): component i of vector y(j). • yˆi (j): component i of vector yˆ(j). Weighting factors (kij ) are used for several purposes:
• To scale the weight of the different inputs. x
• To give more importance to certain strategic timings. Once the cost function is established it is necessary to choose an optimization technique. For complicated models (nonlinearities, saturation, high order, etc.) the optimization problem can be very complicated, and then a powerful optimization technique is required. A Genetic Algorithm is a very good candidate for this role, even more so if there is no restriction on computational cost (it is an off-line identification). Two examples are shown:
q
Y1(w)
w
Wm2 w1
w2
V1
Wm1
V2 dc-motor dc-motor
Figure 1: Conveyor system scheme.
• Parameter identification of a continuous linear model but with a high number of poles. • Parameter identification of a non-linear model.
G2j =
Ki , j = 1...2 τj s + 1
(5)
4 Conveyor system identification The complex aspect of this problem results in the fourth order polynomial in transfer functions G1j . An additional handicap is that the range value of the polynomial coefficient akj is unknown and with these parameters it is difficult to establish a relationship with time response. This problem can be overThe process signals are: come by changing these parameters using the poles of the system pkj (this is easier to relate with time • Inputs: Motor amplifier voltage inputs (V1 and response). V2 ).
The conveyor system prototype (see figure 1) enables practical investigation in a common industrial problem, that is, tension and speed control of transported materials in a process used in the textile and paper industry.
• Outputs: Speed Y1(w) and horizontal displace- 4.1 Experiments ment of the pivoting pulley θ. Experiments are made at the operating point Y1(w) = • Auxiliary outputs: Motor speed WM 1 and WM 2 . 4.2 V olts, θ = 0.2 V olts, V1 = 5 V olts and V2 = 5 V olts. Two step signals are applied at the inputs Although the model is non-linear and multivari- of the system. Input V1 goes from 5 to 6 V olts in able, for showing an application with complex lin- the time interval [11, 20] sec and input V2 goes from ear problems, the original model is linearized, thus 5 to 4 V olts in the time interval [26, 34] sec. Figure 2 shows experimental results. obtaining the following linear model structure [6]: "
Y 1 (w) θ
G1j =
#
=
"
G11 G12 G21 G22
#"
V1 V2
#
K1j a0j s4 + a3j s3 + a2j s2 + a1j s + a0j
The experiment shows a dynamic which is similar to the first order models for G21 and G22 . In this (3) case the process identification is easy and well established [10]. But G11 and G12 are more difficult and so the method with GA will be applied in this (4) case.
Figure 3 shows experimental data used for G11 identification, this data is an extract from figure 2 corresponding to time interval [19, 22.25] sec. and the operation point has been subtracted.
V1
6 5.5 5 0
5
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35
0
5
10
15
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25
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35
V2
5 4.5
0
4 −0.2 V1(v)
Angle
2 1
−0.4 −0.6
0 −0.8
−1
0
5
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15
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25
30
35
−1 0
0.5
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1.5 Time(sec)
2
2.5
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4.5 4 3.5
0
5
10
15
20 Time(sec)
25
30
35 0
Figure 2: Experimental results for G1j identification.
Angle(v)
Y1(w)
5
−0.5
−1
4.2 G1j identification
−1.5
Gains K11 and K12 of G1j are easy to obtain: K11 =
θ θ , K12 = ∆V1 ∆V2
Figure 3: Data for G11 identification. (6) The gain value:
θ −0.57 V olts K11 = = = 0.57 For obtaining the model poles, identification with ∆V1 −1 V olts GA must be used. Cost function for G1j identificaG11 poles are obtained from GA optimization of: tion in the time interval [0, te] (N samples) results: J(ζj ) =
N X
ˆ ktj |θ(t) − θ(t)|
t=1
Where: • N : number of experiment samples. • ktj : weighting factors. • θ(t): experimental displacement value at t. ˆ • θ(t): simulated displacement value at t.
(7)
J(ζ1 ) =
N X
ˆ kt1 |θ(t) − θ(t)|
(8)
t=1
Experimental data shows an oscillatory behaviour, and so for simplicity’s sake complex poles are assumed: s11 , s21 = σ1 ± jwp1 ; s31 , s41 = σ2 ± jwp2 Parameters vector can be set to: ζ1 = (σ1 , wp1 , σ2 , wp2 )
Weighting parameters kt1 are set to one, except in • ζj = (s1j , s2j , s3j , s4j ): parameters vector, several strategic points (marked with arrows in figure which means G1j poles. 4), where the value is kt1 = 106 . In this way, a good fit for the oscillation is achieved. 4 3 2 ˆ For θ(t) simulation, polynomial s + a3j s + a2j s + GA characteristics are: a1j s + a0j is obtained from the system poles (s1j , s2j , s3j , s4j ). Then the G1j transfer function • Real codification chromosome. is converted into state space representation to allow Runge-Kutta integration. • Number of individuals: 400.
0.2
• Ranking operator.
0
• Selection operator: stochastic universal sampling.
−0.4 Angle(v)
• Crossover operator: intermediate recombination with a probability of 0.9.
−0.2
• Mutation operator: random with a Gauss distribution (σ = 0.5%) and a probability of 0.2.
−0.6
−0.8
−1
• Search space: σ1 ∈ [−0.2 . . . − 4.8], σ2 ∈ [−1.0 . . . − 8.0], wp1 ∈ [3.0 . . . 50.0] and wp2 ∈ [40.0 . . . 90.0].
−1.2
−1.4
0
0.5
1
1.5 Time(sec)
2
2.5
3
Results in: Figure 5: Real output θ (continuous line) vs. model s11 , s21 = −1.4 ± j29.85 ; s31 , s41 = −6.9 ± j65.52 output θˆ (dotted line) for step V . Weighting factors 1 Kt1 = 1, ∀t. 6 G11 =
0.57 · 3.87 · 10 s4 − 16.6s3 + 5.27 · 103 s2 − 2.45 · 104 s + 3.87 · 106
Minimization process results (weighting factors marked with arrows in Figure 4 shows simulation results compared to ex- kt2 = 1 except at the points 5 perimental data. Figure 5 shows results with Lt1 = figure 7 where kt2 = 10 ) : 1 ∀t, the second case shows a worse fitting for sims12 , s22 = −0.9±j29.84 ; s32 , s42 = −7.46±j50.00 ulated results. 0.2
Then:
0
G12 =
−0.4 · 2.27 · 106 s4 − 16.72s3 + 3.47 · 103 s2 − 1.79 · 104 s + 2.27 · 106
−0.2
−0.4 Angle(v)
1 0.8 V2(v)
−0.6
0.6 0.4
−0.8
0.2 0 −1
−1.2
0
0.5
1
1.5 Time(sec)
2
2.5
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3.5
4
0
0.5
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1.5
2 Time(sec)
2.5
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3.5
4
0.4
3
0.2
Figure 4: Real output θ (continuous line) vs. model output θˆ (dotted line) for step V1 . Weighting factors Kt1 = 106 at points marked with arrows.
Angle(v)
0 −0.2 −0.4 −0.6 −0.8 −1
For G12 identification the procedure is the same. Figure 6 shows experimental data used (extracted Figure 6: Step in the input V and response of the 2 from 2, [33, 37] sec.). The gain value is: output θ. −0.4 V olts θ K12 = = = −0.4 Real and simulated values are compared in figure 7. ∆V2 1 V olts
0.4
o 2.2 C T (s) = u(s) (200s + 1)(15s + 1) %
0.2
0
But this model has a limited validity range (only near the operating point of the linear model), as shown in figure 9. A solution consists in using a linear multimodel [4], but this is not feasible when the process dynamic depends on many parameters, or when nonlinearity is hard. In these cases, a non-linear model must be calculated.
Angle(v)
−0.2
−0.4
−0.6
−0.8
0.5
1
1.5
2 Time(sec)
2.5
3
3.5
4
Figure 7: Real output θ (continuous line) vs. model output θˆ (dotted line) for step V2 . Weighting factors Kt1 = 105 at points marked with arrows.
100 50 0
Control action u(%)
0
Temperature(ºC)
150 −1
0
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80 60 40 20
5 Thermal process identification The thermal process prototype (figure 8) is used to simulate industrial heating systems. It is composed of a box in which there is a regulated power supply, a resistance, two temperature sensors, and a fan which circulates air. The manipulated variable is the voltage percentage at the power supply input (u). The output variable is the measured temperature inside the resistance (y). The inside temperature is also measured (Ti ). Input Voltage (u)
Measured Resistance Temperature (y)
Regulated Power Supply
Resistance
Temperature Sensor
Resistance Temperature (ºC)
Error(ºC)
20 10 0 −10
Figure 9: Identification of the process using a lineal model. Response in continuous line and model in dotted line. Process input and model error is shown below. From first principles [11] a non-linear structure can be established as: x˙1 = k1 u2 − k2 (x1 − Ti ) 1 x˙2 = (x1 − x2 ) k3 yˆ = x2
Inside Temperature Tin (ºC)
Figure 8: Thermal process prototype scheme. From modelling (using first principles) a set of nonlinear differential equations is obtained. A nonlinear behaviour can be easily verified with a few experiments. It is possible to obtain a linear model by traditional methods:
Where: • yˆ: resistance temperature (o C) • u: voltage percentage at power supply input (%). • Ti : inside temperature (o C). • k1...3 : model parameters to identify.
160
The cost function is: J(k1 , k2 , k3 ) =
te=2300 X
140
|y(j) − yˆ(j)|
j=1
GAs characteristics are the same as before, but the search space: k1 ∈ [0 . . . 0.1], k2 ∈ [0 . . . 0.01] and k3 ∈ [10.0 . . . 25.0].
Temperature(ºC)
120
100
80
60
Figure 10 shows the model simulation and errors with the parameters obtained from GA minimization: k1 = 86 · 10
−6
k2 = 6.027 · 10
−3
0
1000
2000
3000 Time(sec)
4000
5000
6000
Figure 11: Validation of non-linear model. The process response is shown in continuos line and the model in dotted line.
100 Temperature(ºC)
20
k3 = 16.75
120
80 60 40 20
0
500
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2000
0
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1000 Time(sec)
1500
2000
2 1.5 1 Error(ºC)
40
0.5 0 −0.5 −1
This work shows an off-line parameter identification where prior physical information is used. This method uses qualitative and quantitative information from the first principles, and it is completed by parameter identification. This technique is an attractive alternative to methods based on neural networks or similar.
It is noteworthing noting the use of a weighting factor in the conveyor system identification to guide the Figure 10: Identification of non-lineal model using identification process. Another advantage is the low GAs. Process response in continuos line and model sensitivity to noise, this is because the structure is in dotted line. The model error is shown below. previously obtained from modelling and the noise Cost function takes the value 854.2 for 2300 sam- has not been included. ples, which is a mean error of 0.341o C/sample. Finally, there is no condition for input signals in orThis is a small error compared with the quantifica- der to identificate, and this means that the technique tion error of the digital converter which has been can be applied using data from regular process operused (0.18o C). ations. Since it is not necessary to make a specific Validation with other experimental data is shown in experiment the technique is useful for industrial sysfigure 11. Quite a good fit has been obtained even for tems. high and low temperatures. With this data the mean error is 0.92o C/sample. Acknowledgements
6 Conclusions
This work has been partially financed by European FEDER funds, project 1FD97-0974-C02-02.
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