# Dynamic Modeling and Optimal Control for Complex Systems with

Authors: Lingli Guo; Zhengguang Xu; Ye Wang
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## 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 byK-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.


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## Body

## 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 viaK-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 Section2. 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 K-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. Section2 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 Table1, 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 Figure1.Figure 1
Working state time series in feature space.ByK-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 Table2.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
14.1273For 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 (Figure3) 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 (Figure5) by residual independence algorithm and statistical F-test.Figure 5
Residual independence series test.Using the statisticalF-test method, we set the parameter α=5% and gain Fα(∞)=3. Table 3 is from the formula F=(Jn1-Jn2/2n2-n1)×(L-2n2/Jn2).Table 3
Every class number and class centre.
The ordern
Residual sum of squaresJ
F-value
1
314.5495
2
296.1067
13.8894
3
289.0520
5.4182
4
287.7628
0.9901 < 3
5
279.3846
0.1452Finally, the system model can be described as(1)y~k-0.4296yk-1-0.1706yk-2-0.0896yk-3=0.1362uk-1-0.0443uk-2-0.0068uk-3,yk=Fy~k,where y~(k) is the initial output of the system at k instant,y(k) is corresponding class centre, F is classifier, and u(k) 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(2)x1k+1=x2k+h2uk,x2k+1=x3k+h1uk,x3k+1=-a0x1k-a1x2k-a2x3k+h0uk.The output is expressed as(3)yk=x1k+h3uk,where h3=b3, h2=b2-a2h3, h1=b1-a1h2, and h0=b0-a0h3-a1h2-a2h1, ai, bi, and hi are given by the following model:(4)yk+a2yk-1+a1yk-2+a0yk-3=b3uk+b2uk-1+b1uk-2+b0uk-3.In order to implement the control for the system, the discrete-time state space model can be obtained as follows:(5)x1k+1x2k+1x3k+1=0100010.08960.17060.4296x1kx2kx3k+0.1362-0.0211-0.0073uk,yk=100x1kx2kx3k,and (5) can be simplified as(6)Xt+1=AXt+BUt,(7)yt=CXt.Combining the expert knowledge with the above system analysis, the steady-state operating mode centre for reference can be set as(8)r0=0.6650.Now, the steady-state state space model is obtained by (6), (7), and (8):(9)r0=CX0,(10)X0=AX0+Bu0.Therefore, we can get the steady-state part of the control variable and state variable by (10):(11)u0=CI-A-1B-1r0,X0=I-A-1Bu0.The associated cost function is defined by(12)J=∑k=0∞xkTQxk+ukTRuk,where xk=Xk-X0, x(k)∈Rn, is the state of the system at instant k∈N+={0,1,2,…}; u(k)=U(k)-u0, u(k)∈Rm, is input of the control; Q∈Rn∗n and R∈Rm∗m 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 {u^(k),k=0,1,…,N-1} so that the cost function (12) is the minimum.And (12) can be transformed into(13)Jxk=xTkQxk+uTkRuk+∑n=k+1∞xTnQxn+uTkRuk=xTkQxk+uTkRuk+J^xk+1,where J(x0)=0 is required as the boundary condition. Therefore, J(xk) becomes a Lyapunov functional. Based on Bellman’s optimality principle, the cost functional J(xk) becomes invariant and satisfies the discrete-time Hamiltonian for the infinite horizon optimization case, and it becomes(14)J^xk=minukxTkQxk+uTkRuk+J^xk+1.We guess the solution of Bellman’s equation can be expressed as(15)J^xk=xTkGkxk,where G(k)∈Rn∗n is the real symmetric variable matrix.Substituting (14) into (15) yields(16)J^xk=xTkGkxk=minukxTkQxk+uTkRuk+xTk+1Qxk+1.Combining (15) and (6), it follows that(17)J^xk=xTkGkxk=minukxTkQkxk+uTkRuk+Axk+BukTGk+1Axk+Buk=minukuk+FkxkTxTkQk+ATGk+1A-FTkR+BTGk+1BFkxk+uk+FkxkT×R+BTGk+1Buk+Fkxkuk+FkxkT,where (18)Fk=defR+BTGk+1B-1BGK+1A.In (6), the second part is only related with u(k), and also it is easy to prove that R(k)+BTG(k+1)B is positive define matrix. Therefore, the enough and necessary condition for minimizing the cost function J^ is (19)u^k=-Fkxk.The optimal control law is obtained.Substituting (19) into (17) yields (20)J^xk=xTkGkxk=xTkQ+ATGk+1A-FTkR+BTGk+1BFkxk.Considering (17) exists for every x(k), so(21)Gk=Q+ATGk+1A-FTkR+BTGk+1BFk.Substituting (18) into (21) yields the G(k) meeting the following Riccati equation:(22)Gk=Q+ATGk+1A-ATGk+1BR+BTGk+1B-1BGk+1A,where G(N)=Q 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 parametersA,B,R,Q,S of the discrete-time state space model, we build the Riccati equation  (22).(ii)
FromG(N), by reverse recursive method, calculating the GN-1,GN-2,…,G(1),G(0) and then substituting them into (18) and (19) yield u^(k).(iii)
Combiningx(k)=X(k)-X0, u(k)=U(k)-u0 and the initial condition, the optimal control law can be obtained by(23)Uk=-Fxk+FX0+u0.Similarly, the state trajectory is(24)Xk+1=A-BFkXk+BFkX0+Bu0.Substituting (24) into output model yields(25)yt=CXt=x1t.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 Section3.(
1
) The results for the initial state cell [1,1,1] 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.(
2
) The results for the initial state cell [17,17,17] 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 Figures6, 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.



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*Source: 245685-2015-06-09.xml*