This paper is concerned with the problem of the working state class dynamic modeling and control of the complex system with statistical trajectory. Firstly, a novel discrete-time nonlinear working state class dynamic mathematical model of sintering machine is constructed by
In control engineering, many techniques for designing a controller, either mathematical analysis or numerical calculations [
However, we are inspired by workers keeping watch on the fire who recognize the working state by observing the flame type to adjust the firing temperature. To some extent, parameters of the working state such as bellows temperature and bellows pressure which can be measured reflect flame type or working state; it is quite natural to inquire whether we imitate the whole manual control to automate control in order to avoid artificial factors. Dynamic model of the working state classes was built via dividing these parameters values into interval number [
In this paper, we show that cell structured state spaces modeling gives us indeed a new possibility. In particular, we present controlled autoregressive moving average (CARMA) modeling method of the dynamics working state class for sintering machine, which is established by dividing state space into cells via
In this paper, we proposed the modeling method that the system working class dynamic behavior is completely recognized by identifying the system cell dynamics. This method described in greater detail is in Section
The optimal control object is the systems that enter into the target cell representing the best working state from any initial cell with the minimum cost function. Section
In this section, we construct the dynamic model of the working state class for the sintering machine through data driven approach. As it is known, there are complex physical and chemical changes so that the mechanism modeling is very difficult. Meanwhile, the operating mode is only estimated according to the set of some approximate variables values such as dynamic west bellows temperature and pressure and east bellows temperature and pressure. Therefore, the first step is clustering dynamic variables values. The data must be standardized before clustering. The part of sample data is shown in Table
The partial data in parameter space.
WBT20 | WBT21 | WBT22 | EBT20 | EBT21 | EBT22 |
---|---|---|---|---|---|
419 | 431 | 306 | 401 | 382 | 352 |
321 | 425 | 329 | 403 | 394 | 353 |
440 | 449 | 324 | 410 | 387 | 361 |
371 | 399 | 262 | 418 | 383 | 352 |
442 | 475 | 290 | 382 | 388 | 382 |
492 | 452 | 309 | 338 | 387 | 283 |
401 | 370 | 366 | 437 | 456 | 297 |
411 | 378 | 357 | 432 | 441 | 374 |
398 | 384 | 364 | 421 | 434 | 379 |
421 | 394 | 394 | 424 | 429 | 392 |
421 | 392 | 384 | 411 | 434 | 369 |
400 | 374 | 365 | 411 | 432 | 374 |
435 | 408 | 385 | 380 | 400 | 300 |
420 | 404 | 375 | 390 | 421 | 270 |
400 | 409 | 378 | 348 | 430 | 268 |
415 | 409 | 373 | 350 | 415 | 264 |
390 | 378 | 370 | 408 | 438 | 268 |
390 | 390 | 364 | 420 | 430 | 310 |
433 | 391 | 369 | 454 | 469 | 306 |
⋯ | ⋯ | ⋯ | ⋯ | ⋯ | ⋯ |
The standardized data sequence is in Figure
Working state time series in feature space.
By
JC values change with category numbers.
The total clustering centers are shown in Table
Every class number and class centre.
Class number | Class centre |
---|---|
1 | −20.4331 |
2 | −4.9354 |
3 | −3.3971 |
4 | −2.4656 |
5 | −1.7454 |
6 | −1.1826 |
7 | −0.6749 |
8 | −0.2130 |
9 | 0.2429 |
10 | 0.6650 |
11 | 1.1314 |
12 | 1.6472 |
13 | 2.2207 |
14 | 2.9349 |
15 | 4.0309 |
16 | 6.5470 |
17 | 14.1273 |
For describing the working state class trajectory, the second step is obtaining every instant mode by the clustering centre instead of each element belonging to this class in order to build CARMA model. Firing temperature is the input of the model. Clustering centre is the model output. The order (Figure
Residual sum function of squares change with the order.
Residual sum function of squares change with the delay.
The model parameter is estimated by least square method. Then, the dynamic mode model is tested (Figure
Residual independence series test.
Using the statistical
Every class number and class centre.
The order |
Residual sum of squares |
|
---|---|---|
1 | 314.5495 | |
2 | 296.1067 | 13.8894 |
3 | 289.0520 | 5.4182 |
4 | 287.7628 | 0.9901 < 3 |
5 | 279.3846 | 0.1452 |
Finally, the system model can be described as
The controller is designed to guide the sintering machine working state class to follow a predefined class trajectory. To obtain the control of the system, we assume the state variable is
In order to implement the control for the system, the discrete-time state space model can be obtained as follows:
and (
Combining the expert knowledge with the above system analysis, the steady-state operating mode centre for reference can be set as
Now, the steady-state state space model is obtained by (
Therefore, we can get the steady-state part of the control variable and state variable by (
The associated cost function is defined by
And (
We guess the solution of Bellman’s equation can be expressed as
Substituting (
Combining (
In (
The optimal control law is obtained.
Substituting (
Considering (
Substituting (
As discussed above, the discrete-time optimal control problem algorithm can be established as follows. According to the parameters From Combining
Similarly, the state trajectory is
Substituting (
Meanwhile, due to the one-to-one mapping of the state space and cell space, we can gain the corresponding the state cell trajectory and output cell trajectory.
However, every cell is possibly the initial cell in sinter machine working state class, so we can obtain a control law for every cell. By (
The controller design is completed at this point.
The performances of the proposed optimal controller are illustrated by simulation. The principle of simulation setup based on the experimental schematic is described by Section
Initial system output trajectory.
Finial system output trajectory.
System output cell trajectory.
System input trajectory.
Initial system output trajectory.
Finial system output trajectory.
System output cell trajectory.
System input trajectory.
Dynamic tracking of the sintering machine working state class response is shown in Figures
The ignition temperature is a key factor affecting the sinter performance; thus its automation control contributes to maintaining good quality. In view of this idea, this paper has proposed a method to model cell dynamics describing the working state class for a class of complex systems with statistical trajectory characteristics. Based on the dynamic programming optimal algorithm, the control law is constructed for each initial cell. The model method and optimal control strategy have important advantages in terms of state classification by cell division and dynamic class description. Their weaknesses (such as the cell boundary and stability) are constantly improved. This unconventional approach to describe the class trajectory is probably the tip of the iceberg in the field of the working state class modeling and control that will emerge in the coming years. A truly multidisciplinary research approach, where all the aspects of such complex dynamic systems have to be considered and fully understood, is required to fully exploit the potential of this new branch of applied science.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work was supported by the National Natural Science Foundation of China (Grant no. 61175122) and Key Laboratory of Innovation Method and Decision Management System of Guangdong Province, Guangzhou, Guangdong (Grant no. 2011A060901001-14D). Professor Xu is the first author’s tutor for a doctor’s degree. His idea that pattern trajectory describes the complex systems moving influences the first author deeply. But he does not point to the specific methods how to characterize the pattern and control the pattern trajectory.