Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 245685, 8 pages
http://dx.doi.org/10.1155/2015/245685

Research Article

Dynamic Modeling and Optimal Control for Complex Systems with Statistical Trajectory

Lingli Guo,1,2 Zhengguang Xu,¹ and Ye Wang¹

¹School of Automation, University of Science & Technology Beijing, Beijing 100083, China
²Department of Mathematics, Changzhi College, Changzhi, Shanxi 046011, China

Received 29 June 2014; Revised 31 August 2014; Accepted 31 August 2014

Academic Editor: Qingang Xiong

Copyright © 2015 Lingli Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 -means cluster method for the local data, which overcomes difficulties of modeling by mechanism and describes the class change against the point movement. Then, based on the working state class dynamic modeling, the optimal control method which is significant to achieve trajectory tracking for sintering machine is developed by constructing the quadratic performance indicator. More importantly, the method facilitates control process realization from one class to another. Finally, the simulation is provided to illustrate the effectiveness of the proposed method.

1. Introduction

In control engineering, many techniques for designing a controller, either mathematical analysis or numerical calculations [1–4], are based on the mathematical model of the system. However, if the system is nonlinear or the model of the system is not clearly defined, it is a difficult task to design a controller that satisfies the system requirements. For example, there are complex physical and chemical transformations in sintering machine so that they are difficult to model by the physical nature.

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 [5–7]. Predictive control model was established by the class centre value representing the whole class points [8–11]. While these models describe the dynamic moving of flame type or working state class, the class representation is still thought to be significant challenge for working state computation.

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 -means [9] clustering algorithm. Based on this model, the dynamic programming optimal control strategy is employed to design controller for every initial cell. In spite of optimal control theory based on cell mapping and its various areas of application in engineering, traditional methods [3–12] are known model, and the control table is obtained by cell mapping training. In [13], although the model is unknown, the order is known, and the system dynamic and control table is gained by learning signal.

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 2. In [14–17], the pattern recognition is static process. Therefore, the dynamics of the system are then described in the form of mappings from one cell onto another in this paper. In addition, most of the cell division principles are based on the experience [13, 18–22]. Here, we present the cell division by -means cluster method. The initial category and category center of system working state randomly are selected until the JC [3] value tends to make steady. Meanwhile, combining with the experience, the best clustering result finally is decided. Every cell represents a working state class, so cell-to-cell mapping describes the working state class dynamics. References [23–26] involved dynamic system simulation.

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 2 contains the constructing process of the model. The attractiveness of the method lies in eliminating the division error at the experience level. Section 3 is devoted to introducing the dynamical planning optimal control strategy. Example and simulation of applying the extended method to sinter machine control are finished in Section 4. Section 5 concludes the paper. Emphasis is placed on the construction of dynamic working state model by which the class shift with tracking statistical motion trajectory is evaluated.

2. Problem Statement and Dynamic Description of the Sintering Machine Working State Class

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 1, where the west bellows temperature is simplified as WBT and east bellows temperature is EBT.

Table 1: 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 1.

Figure 1: Working state time series in feature space.

By -means clustering and JC rules (Figure 2), we find the best clustering number is 17.

Figure 2: JC values change with category numbers.

The total clustering centers are shown in Table 2.

Table 2: 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 3) and delay (Figure 4) of the model are decided by the inflection point of residual sum function of squares.

Figure 3: Residual sum function of squares change with the order.

Figure 4: 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 5) by residual independence algorithm and statistical -test.

Figure 5: Residual independence series test.

Using the statistical -test method, we set the parameter and gain . Table 3 is from the formula .

Table 3: Every class number and class centre.

The order
Residual sum of squares
-value

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

where is the initial output of the system at instant, is corresponding class centre, is classifier, and is the input.

3. Dynamical Optimal Controller Design

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

The output is expressed as

where , , , and , , , and are given by the following model:

In order to implement the control for the system, the discrete-time state space model can be obtained as follows:

and (5) can be simplified as

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 (6), (7), and (8):

Therefore, we can get the steady-state part of the control variable and state variable by (10):

The associated cost function is defined by

where , , is the state of the system at instant ; , , is input of the control; and are the real symmetric positive definite weighting matrix. In this paper, we assume that all the states are measurable and available. The controller object is finding the controller series so that the cost function (12) is the minimum.

And (12) can be transformed into

where is required as the boundary condition. Therefore, becomes a Lyapunov functional. Based on Bellman’s optimality principle, the cost functional becomes invariant and satisfies the discrete-time Hamiltonian for the infinite horizon optimization case, and it becomes

We guess the solution of Bellman’s equation can be expressed as

where is the real symmetric variable matrix.

Substituting (14) into (15) yields

Combining (15) and (6), it follows that

where

In (6), the second part is only related with , and also it is easy to prove that is positive define matrix. Therefore, the enough and necessary condition for minimizing the cost function is

The optimal control law is obtained.

Substituting (19) into (17) yields

Considering (17) exists for every , so

Substituting (18) into (21) yields the meeting the following Riccati equation:

where is required as the boundary condition.

As discussed above, the discrete-time optimal control problem algorithm can be established as follows.

(i)According to the parameters of the discrete-time state space model, we build the Riccati equation  (22).(ii)From , by reverse recursive method, calculating the and then substituting them into (18) and (19) yield .(iii)Combining , and the initial condition, the optimal control law can be obtained by

Similarly, the state trajectory is

Substituting (24) into output model yields

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 (23), the controller is the best so that the cost function is optimal.

The controller design is completed at this point.

4. Numerical Simulations

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 3.

The results for the initial state cell are shown in Figures 6–9.

Figure 6: Initial system output trajectory.

Figure 7: Finial system output trajectory.

Figure 8: System output cell trajectory.

Figure 9: System input trajectory.

The results for the initial state cell are shown in Figures 10–13.

Figure 10: Initial system output trajectory.

Figure 11: Finial system output trajectory.

Figure 12: System output cell trajectory.

Figure 13: System input trajectory.

Dynamic tracking of the sintering machine working state class response is shown in Figures 6, 7, 8, 10, 11, and 12. Since the best working state class is the number 9 cell, its state reference is set to the cell centre 0.6650. From these figures, it is obvious that the optimal controller can realize zero steady-state error and has good dynamic and steady response characteristics.

5. Conclusion

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.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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.

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