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Abstract—This paper presents some applications of a Model Based Predictive Control (MBPC) type algorithm which is applied to nonlinear processes. The basic idea of the algorithm is the on-line simulation of the future behaviour of the control system, by using a few candidate control sequences. Then, using rule based control these simulations are used to obtain the ‘optimal’ control signal. The efficiency and applicability of the proposed algorithm are demonstrated through applications.

I. INTRODUCTION

ODAY, many industrial systems are still controlled by simple PID (proportional-integrative-derivative)

algorithms, despite the better performances usually provided by systems developed following the modern control theory. PID controllers can be used to control a wide range of different processes, need only rough process models to be easily tuned and give pretty good set-point tracking performances. On the other hand it is clear that PID performances, although satisfactory, could be improved when dealing with highly nonlinear processes, or processes featuring unmodeled dynamics and external disturbances.

A Model Based Predictive Control (MBPC) algorithm is described by using a model to compute the predicted process outputs. The parameters of the model are obtained through an identification algorithm. Also, a cost function related to the closed loop performance of the system is defined, and the control signal is obtained by means of minimization the cost function. Finally, the first of these signals is applied to the process [1]. The extension of linear MBPC to nonlinear processes is straightforward at least conceptually. But there exists some difficulties [2]: the availability of nonlinear models due to the lack of identification techniques for nonlinear processes, the computational complexities, the lack of stability and robustness results.

Manuscript received January 31, 2007. R. Bălan is with Technical University of Cluj-Napoca, Dept. of

Mechanisms, Precision Mechanics and Mechatronics, 103-105, B-dul Muncii, 400641, Cluj-Napoca, Romania (e-mail: radubalan@yahoo.com).

V. Mătieş is with the Technical University of Cluj-Napoca, Dept. of Mechanisms, Precision Mechanics and Mechatronics, 103-105, B-dul Muncii, 400641, Cluj-Napoca, Romania (e-mail: vistrianmaties@yahoo.com)

O. Hancu is with Technical University of Cluj-Napoca, Dept. of Mechanisms, Precision Mechanics and Mechatronics, 103-105, B-dul Muncii, 400641, Cluj-Napoca, Romania (e-mail: ohancu@netscape.net).

S. Stan is with the Technical University of Cluj-Napoca, Dept. of Mechanics and Programming, 103-105, B-dul Muncii, 400641, Cluj-Napoca, Romania (40264-401684; e-mail: sergiustan@ hotmail.com).

C. Lăpuşan is with Technical University of Cluj-Napoca, Dept. of Mechanisms, Precision Mechanics and Mechatronics, 103-105, B-dul Muncii, 400641, Cluj-Napoca, Romania (e-mail: lapusanciprian@yahoo.com).

The purpose of the controller is typically to force the output to follow the reference signal. If reference is a constant, the problem is commonly referred to as set-point regulation. When the reference is time varying (but is known in advance), defining a control law to force the output to follow the reference signal is called the positioning control.

This paper presents some applications of a MBPC type algorithm. The basic idea of the algorithm is the on-line simulation of the future behavior of the control system, by using a few candidate control sequences [3]. Then, using rule based control, these simulations are used to obtain the ‘optimal’ control signal.

II. SET-POINT REGULATION ALGORITHM (MBPC-A1)

In [4] it was proposed an algorithm (MBPC-A1) designed for setpoint regulation problem (but setpoint can be arbitrary changed). The main idea of the algorithm is to compute for every sample period:

- the predictions of output over a finite horizon (N);- the cost of the objective function, for all (theoretically case) or a few (practically case)

possible control sequences: � � � �)(),..,1(),(. Ntututuu ���and than to choose the first element of the optimal control

sequence. For a first look, the advantages of the proposed algorithm include the following:

-the minimum of objective function is global; -this algorithm can be easy applied to nonlinear processes; -the constraints can easily be implemented. The drawback of this scheme is a very long computational

time, because there are possible a lot of sequences. Therefore, the number of sequences must be reduced. For a first stage, there were proposed [4] the next four control sequences:

� � � �minminmin1 ,..,, uuutu �� � � �minminmax2 ,..,, uuutu � (1) � � � �maxmaxmin3 ,..,, uuutu �� � � �maxmaxmax4 ,..,, uuutu �

where umin and umax are the accepted limits of the control signal, limits imposed by the practical constraints. These values can depend on context and can be functions of time. Using these sequences results four output sequences y1(t),y2(t), y3(t), y4(t). The control signal is computed using a set of rules based on the extreme values ymax0, ymax1, ymin0, ymin1

(fig. 1- d is dead time, t1=N, yr is setpoint) of the output predictions. In the followings, considering processes with positive sign, it can be put in evidence four usual cases:

Nonlinear Control Using a Model Based Predictive Control Algorithm

Radu Bălan, Vistrian Mătieş, Olimpiu Hancu, Sergiu Stan, Lăpuşan Ciprian

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Proceedings of the 2007 IEEE International Symposium onComputational Intelligence in Robotics and AutomationJacksonville, FL, USA, June 20-23, 2007

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Case 1: If ymax0<yr (corresponding to u1(t) sequence) and ymax1>yr (corresponding to u2(t) sequence) Then (using a linear interpolation):

0max1max

0maxmax1maxmin

0max1max

minmax)(yy

yuyuy

yyuu

tu r

�

� (2)

Case 2: If ymin0<yr (corresponding to u3(t) sequence) and ymin1>yr (corresponding to u4(t) sequence) Then (using a linear interpolation):

0min1min

0minmax1minmin

0min1min

minmax)(yy

yuyuyyyuutu r

�

� (3)

Case 3: If: ymax0>yr Then u(t0)=umin

(4) Case 4: If: ymax1<yr Then u(t0)=umax

(5) In fig. 1, every output prediction curve is marked with a

number which correspond to the number of control sequence from relations (3). Similar to case 3 and case 4, there are two similarly cases if dy/dt<0 for t<t0.

If the algorithm uses only these 6 rules, the variance of u(t) will be large [4]. So, in the second stage, depended by behaviour of the control system, are used next methods:

-an algorithm that modifies the limits of control signal:

umin ≤ uminst(t) ≤ u(t) ≤ umaxst(t) ≤ umax ; Δumin≤ Δu≤ Δumax (6)

Fig. 1. Examples of output predictions For example:

� � � � � � � � � �� �tytytutuftu r,,1,1 maxstminst1minst � (7) � � � � � � � � � �� �tytytutuftu r,,1,1 maxstminst2maxst � (8)

where f1, f2 are functions which decrease or increase (depended by behavior of the control system) the difference between umaxst(t) and uminst(t). In relations (1).. (5), the values of umax, umin are replaced with uminst(t), umaxst(t). In the following, if is necessary, the next relations are used:

� � � � � �� �11 minminstminst �� tuuktutu ststst (9) � � � � � �� �ststst utuktutu � 11 maxmaxstmaxst (10)

where kst is a weight parameter and ust is the estimated value of control signal in steady state. But in some circumstancing (perturbations, inaccurate model) the limits of control signal must increase. Also, it is necessary to limit the minimum value of umaxst(t)-uminst(t)>dust>0, where dust is a parameter of the control algorithm.

-using the “variable setpoint“ [4]:

yr1(t)=yr(t)+kref[y(t)-yr(t)] (11)

where kref is a weight factor - using a filter to compute control signal (especially in

steady state regime). Applications of MBPC-A1 algorithmLet’s consider a linear process (P1):

)3()2()1()3()2()1()(

321

321

dtubdtubdtubtyatyatyaty

����� (12)

where y[.] is the process output, u[.] is the controller output, 0 u[.] 250, A[.]=[1 -2.43492 1.97629 -0.53468], B[.]=[0.000948003 0.004438182 0.001296496], static gain is k0=1; d=1 is dead time.

Example 1 In this example, yr[0]=0 and yr[t]=150 for t>0, u[0]=0 for

t0 and u[t]=150 for t>0. For PID tuning, it is used Ziegler-Nichols criterion (fig. 2). This example shows the advantages of MBPC-A1 algorithm, comparatively with PID: a shorter time response, no override.

Fig. 2. Example 1

Remark: In the next figures, there are represented two or more functions versus sample point, so both axis label only with units. Only the setpoint (SP-yr(t)) and process’s output (PV- y(t)) are represented at true scale. Controller’s output (OP- u(t)) is represented as u(t)/3.

Example 2 In this example (fig. 3) the setpoint has a variable shape,

the model is accurate (non-adaptive case), kref=0.2, dust=5, it is not used information about setpoint changes.

Fig. 3. Example 2 This example shows the effect of parameter kst.Example 3 Conditions: analogous with example 2 but kst=0.0. This

example shows the effect of parameter kref. In this case (fig. 4), the time response is minim, but the variance of u(t) is larger.

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Example 4 Conditions: analogous with example 3 but kst=0.1. This

example shows the effect of the value of kref if kst� 0.0.In this case, in steady state, the difference umaxst(t)-umnist(t)

will decreases (Fig. 5).

Fig. 4: Example 3

Fig. 5. Example 4 Example 5 Conditions: kst=0.05, kref=0.1. In this example, a simple

filter is used to reduce the variance of control signal in steady state regime, using next relation: � � � � � � stuu uktuktu ��� 1 where ust is the estimated value of

control signal in steady state regime and ku is the filter parameter (fig. 6).

Fig. 6. Example 5

Example 6 In this example the static gain (k0) of the process is

variable from 1 to 1.6 with 0.2 step. For identification, it is used a recursive least square algorithm. For control algorithm it is used model-based predictive control (MBPC-A1).

The estimate of static gain is k0est. The forgetting factor is λ=0.98, kst=0.15, kref=0.2, noise: σ =0. If difference between process and model is quite larger, the control algorithm will compute a wrong control signal and it is possible to appear significant errors (for example at step 498 the override is 16% - fig. 7). A method to reduce this effect is to choose

cautions value for parameters, especially for kst and kref.

Fig. 7. Example 6

III. POSITIONING CONTROL ALGORITHM (MBPC-A2)

For positioning control, it is used a specific algorithm. In this case, the rules (2)…(5) can not be applied directly. In fig. 8, are represented the evolutions of errors ei(t)i=1...4versus sample time. Every output prediction curve is marked with a number which correspond to the number of control sequence from relations (3). Notations: t0 is current time, Nis the horizon of output, δ is a parameter which is used for a fine-tuning (first, it is more simple to consider δ=0).

Fig. 8. Output predictions (Cases 1..4)

It is used next five rules: Case 1: The sequence u3(t) leads to:

� �� �teNtdt 300 minmin

���� , ��0min (13)

In this case u(t)=uminst(t). Case 2: The sequence u2(t) leads to:

� �� �teNtdt

20

1 maxmax���

� , ��1max (14)

In this case u(t)=umaxst(t).Case 3: The sequence u4(t) leads to:

� �� �teNtdt 401 minmin

���� , ��1min and: e4(t0+d+1)>0. (15)

In this case u(t)=umaxst(t).Case 4: The sequence u1(t) leads to:

� �� �teNtdt

10

0 maxmax���

� , ��0max and: e1(t0+d+1)<0. 16)

In this case u(t)=uminst(t). Case 5: In majority of the other situations, the predictions

(for u2 and u3 sequences) are obtained like in fig.9. In this case it is used a linear interpolation:

� � � � � �01

0maxst1minst

minmaxminmax

��

�tututu (17)

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Fig. 9. Output predictions

If the algorithm uses only these 5 rules, the variance of u(t) will be large. There are some solutions to reduce this variation. One of them is to use an algorithm that modifies the limits of control signal based on relations (6). As a result, the difference between umaxst and uminst decreases. On the other hand, in some cases, it is necessary to limit or to increases this difference.

A good behaviour of the control algorithm leads to a prevalence of case 5.

In examples 7, 8 the setpoint has a trapeze or sine shape. MBPC-A2 uses information about future setpoint changes. Controller’s output (OP- u(t)) is represented as 15+u(t)/3; curve (1) is 10·(yr(t)-y (t)). The process is P1.

The static gain k0 is time dependent; it rises from 1 to 2 in 360 steps and then falls from 2 to 1 in 360 steps; the estimation of k0 is k0est. Other parametres: λ=0.94, kst=0.2, σ: 0.4*10-3; 1.0*10-3; 1.6*10-3; 2.2*10-3 (curve (2)). The model has 5 poles and 4 zeros, so the noise can be included in model.

Fig. 10. Example 7. Setpoint: trapeze shape

Fig. 11. Example 8. Setpoint: sine shape

IV. NONLINEAR CONTROL APPLICATIONS

The algorithms presented in previous sections can be directly applied to smooth nonlinear processes. But in other cases, for example if the sign of the process is changing, these algorithms must be particularized.

There are some well-known nonlinear control system design techniques: Lyapunov control design, input-output

linearizing control design, input-state linearizing control design and integrator backstepping control design. In [5] are presented some examples where these methods failed and it's proposed a hybrid method as an alternative nonlinear control system design method. In the following will be used these examples to test the algorithms presented in previous sections. There will be denoted with (P) – the case of MBPC algorithm and with (H) – the case of the hybrid algorithm.

Example 9 Consider the system:

� � � �� �

1

22212

121

15.10

00tanh

xyxuxxx

xuxx

�����

���

�

�

(18)

Using an accurate model, the results obtained by two methods are similar. Though, due to the fact that (P) actions in first place on the state x1 (the output y), the results obtained for the output signal are better. In fig. 12 at step 45 it is noticed the four predictions of the output signal.

Fig. 12. Example 12. Accurate model.

Also, some tests of robustness were realized. In the first test (fig. 13) the control signal u(t) is replaced in equations (18) with 0.5·u(t), in the second test (fig. 14) with 2·u(t).

Fig. 13. Example 12. Robustness test 1.

Fig. 14. Example 12. Robustness test 2. In the first test, (H) becomes unstable, while (P) succeeds

to stabilize the system.

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In the second test both algorithms succeed to stabilize the system. The dominant nonlinearity is the quadratic term 2

2xin the second state equation. Let us consider this equation under the form: uxaxx ���� 2

212� . For a=2.1 (H) is still stable but for a=2.2 (H) becomes unstable. (P) succeeds to stabilize the system even for a=3 with the condition of increasing the limits of the control signals (umax=4, umin=-4).

In fig. 15 is presented the behaviour of (P) in the case of the positioning system, using MBPC-A2 algorithm. The output y (state x1) follows a trapeze reference. It is used an accurate model. This is another example which shows that it is possible to obtain better results using a strategy which uses a bank of controllers and a strategy to switch between them. This example shows that the control algorithm has a good behavior even for positioning case.

Fig. 15. Example 9. Positioning case.

Example 10. Consider the system: � �� �

1

22212

122211

00

12.00

xyxuxxx

xxxxx

�����

����

�

�

(19)

If the model is accurate, the output reply is better in the case (P), but the variance of the control signal and the variance of the state x2 increases (fig. 16). For robustness test, the second equation from (19) is modified:

uaxxx ���� 2212�

If a=0.8 (H) becomes unstable while (P) has a good behaviour (fig. 17).

Fig. 16. Example 10.

To observing the effect of noise it is considered the measured value of the output signal under the form:

� �� �2550001.01 ��� randomxy . It is noticed that for (P) the output signal it follows much better the reference (fig. 18), with a larger variation of state x2.

In fig. 19 it is presented the behaviour of (P) in the case of a positioning system. The output y=x1 follows a trapeze reference. It is noticed that for yr(t)>0.25 the setpoint can not be followed because equation 0112

22 ��� xxxx � does not

have a real solution if 25.011 � xx � .

Fig. 17. Example 10. Robustness test 1.

Fig. 18. Example 10. Noise test.

Fig. 19. Example 11. Positioning case.

Example 11. Consider the system:

� � � �� �

1

22

121

00100sin

xyxuxxxx

�����

�

�

(20)

In the case of the model based predictive algorithm, for examples 11, 12 it is necessary the introduction of new control sequences or/and some supplementary rules. The reason is the fact that system may change its sign. Possible solutions:

-approximation of the actual sign of the system; if the sign is negative, are defined supplementary rules but similar to the rules defined for positive sign.

-usage of some supplementary sequences [6], [7]: � � � �minminmin5 uk,..,uk,uktu ����� � � �0..,,0,0tu 6 � (21) � � � �maxmaxmax7 uk,..,uk,uktu ����

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where k <1 is a parameter of the control algorithm. In the case of usage of accurate model (fig. 20), the reply is more rapid in case of (P). In figure are represented also the form of predictions to the sampling steps 1, 5, 20, 100, 120. Used notations: (a) correspond to the sequences u3 and u4, (b) correspond to the sequences u1 and u2, (c), (d), (e) correspond to the sequences u5, u6, u7, (a1) and (b1) correspond to the sequences u3, u1 respectively u4, u2. It was used k=0.2.

Comparatively with examples 9 and 10 it was produced supplementary rules that permit the choosing of the most rapid way to the reference. For example, for sampling steps 1 and 5 it is chosen u(t)=uminst but for the sampling step 20 it is chosen u(t)=0. For the study of robustness, the equation

ux �2� was replaced by ux �� 22� respectively ux �� 5.02� .In both cases, the hybrid system is very small affected. The predictive algorithm is very small affected in the first case but in the second case appear dumping oscillations. Let us consider now the first equation under the form � �21 5.0sin xx ��� .It can be noticed the sensible increasing of the response time and, in case of MBPC algorithm a certain trend of oscillation of the control signal in the moment in which the error tends to zero.

Another test was realized modifying the initial state. For x1(0)=15 the hybrid system diverges; the predictive algorithm having a good behaviour.

Fig. 20. Example 11.

Example 12. Consider the system: � �

� � � �� �

1

33

2322

121

0000sin

75.30

xyxuxxxxxxxx

��������

�

�

�

(22)

Fig. 21. Example 12.

The behavior of the two algorithms is similar (fig. 21). In the case of (P) algorithm, using of the relations (6) hasn’t lead to favorable behavior

V. CONCLUSIONS

The paper presents a simple and intuitive algorithm applied in the case of some nonlinear process. Using the process model and a reduce number of the sequences control, it is simulated the future behaviour of the process and based on a set of rules it is chosen the signal control considered optimum at the actual moment. Of course there are some difficulties such as the proof of the stability, the way of choosing of the control sequences and the set of rules which will lead to a better result, choosing some parameters etc. Although, taking into account the simplicity of this algorithm the obtained results in the case of the presented examples by nonlinear systems are remarkable. A demo application that implements the proposed algorithm can be downloaded from the last reference.

In the future, starting from the proposed algorithm, the work will focus on: the optimal chosen of the control parameters, the study of other set of control sequences, the study of other set of control rules, adaptive case and practical implementation.

ACKNOWLEDGMENT

This work was supported by the Ministry of Education and Research grant CEEX 112 INFOSOC program. Grant’s title: Simulation, Control and Testing Platform with applications in Mechatronics.

REFERENCES

[1] Camacho E. Bordons, C. (1999) “Model Predictive Control” Spriger-Verlag, ISBN 3-540-76241-8, 1999

[2] Hangos, K.M.; Bokor J., Szederkeny G. (2004), “Analysis and control of nonlinear process systems”, Springer Verlag ISBN 1-85233-600-5, 2004

[3] Bălan R., Mătieş V., Hancu O. (2004) “Model predictive control of nonlinear processes using on-line simulation” in Proceedings of International Conference on Automation, Quality and Testing, Robotics, AQTR 2004-Theta 14, Cluj-Napoca, Romania, 13-15 May 2004, pp. 201-207

[4] Bălan R., Mătieş V., Hodor V., Zamfira I. (2002) “Some issues in the design of adaptive-predictive controllers based on on-line simulation”, International Conference OPTIM 2002, Brasov, Romania, pp. 447-452 ISBN 973-635-012-6

[5] Simmons A., Hung J., Scottedward Hodel A. (2005) “A hybrid improvement to traditional nonlinear control”, IEEE ISIE 2005, June 20-23, 2005, Dubrovnik, Croatia, pag. 49-56, ISBN 0-7803-8739-2, 2005.

[6] Bălan, R., Mătieş, V. Stan, S. (2005) “A Solution of the Inverse Pendulum on a Cart Problem Using Predictive Control”, IEEE ISIE 2005, June 20-23, 2005, Dubrovnik, Croatia, pag. 63-68, ISBN 0-7803-8739-2, IEEE Catalog Number: 05TH8778C

[7] Balan R., Maties, V., Hancu, O., Stan S. (2005) “A Predictive Control Approach for the Inverse Pendulum on a Cart Problem”, IEEE-ICMA 2005 pp. 2026-2031, July 29 - August 1, Niagara Falls, Ontario, Canada, 2005.

[8] Available: http://zeus.east.utcluj.ro/mec/mmfm/download.htm

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