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Influence of control horizon to control quality in predictive control.

1. INTRODUCTION

Predictive control is one of the methods of control design, which has obtained very popular in recent years. Predictive control is based on the use of processes discrete models, and therefore the derivation of the algorithm is implemented mainly in discrete area.

2. PREDICTIVE CONTROL

The term Model Predictive Control presents a class of control methods which have common particular attributes (Camacho & Bordons, 2004; Mikles & Fikar, 2007; Bobal et al., 2009; Bobal, 2009):

* Mathematical model of a systems control is used for prediction of future control of a system output.

* The input reference trajectory in the future is known.

* A computation of the future control sequence includes minimization of an appropriate objective function (usually quadratic one) with the future trajectories of control increments and control errors.

* Only the first element of the control sequence is applied and the whole procedure of the objective function minimization is repeated in the next sampling period.

The principle of Model Based Predictive Control (MBPC) is shown in Fig. 1, where u(t) is the manipulated variable, y(t) is the process output and is the reference signal, and are called minimum, maximum and control horizon. This principle is possible to define as follows:

[FIGURE 1 OMITTED]

1. The process model is used to prediction of future outputs [??] (t) over maximum horizon [N.sub.2]. The predictions are calculated based on information up to time and on the future control actions that are to be determined.

2. The future control trajectory is calculated as a solution of an optimisation problem consisting of objective function and possibility some constraints. The cost function comprises future output predictions, future reference trajectory, and future actions.

3. Although the whole future control trajectory was calculated in the previous step, only first element is actually applied to the process. At the next sampling time, the procedure is repeated. This is known as the Receding Horizon concept.

In this paper the Generalized Prediction Method (GPC) will be used (Clarke et al., 1987).

3. ALGORITHM DERIVATION

Discrete transfer function of second order system has the form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

CARIMA model has the form:

[DELTA]A([z.sup.-1])y(k) = B([z.sup.-1])u(k) + C([z.sup.-1])[e.sub.s](k) (2)

where and are control and output sequence control system, [e.sub.s](k) is white noise. and are polynomials. When polynomial C ([z.sup.-1] = l and [e.sub.s] (k) = 0 then equation (2) describes deterministic system. Substituting individual polynomials from equation (1) to equation (2), the CARIMA model has the following form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

For a general calculation of predictive control for second-order system, must be calculated the first three equations of predicted output.

[??](k + 1) = (1 - [a.sub.2])y(k) + ([a.sub.1] - [a.sub.2])y(k - 1) + +[a.sub.2]y(k - 2) + [b.sub.1][DELTA]u(k) + [b.sub.2][DELTA]u(k - 1) (4)

[??](k + 2) = (1 - [a.sub.1])[??](k) + ([a.sub.1] - [a.sub.2])y(k) + +[a.sub.2]y(k - 1) + [b.sub.1][DELTA]u(k + 1) + [b.sub.2][DELTA]u(k) (5)

[??](k + 3) = (1 - [a.sub.1])[??](k + 2) + ([a.sub.1] - [a.sub.2])y(k + 1) + +[a.sub.2]y(k - 2) + [b.sub.1][DELTA]u(k + 2) + [b.sub.2][DELTA]u(k + 1) (6)

Iteratively substituting equations (3) to (4), and (3), (4) to (5), and (3), (4), (5) to (6) were obtained resulting equations of predicted output through which can be compiled matrix of free and forced system responses.

4. SIMULATION

For simulation was used software The MathWorks[TM] Matlab and Matlab Simulink[R].

In the first step was created initialization script, where was set parameters of system [a.sub.1], [a.sub.2], [b.sub.1], [b.sub.2] and sample time [T.sub.0]. Further was set individual horizons [N.sub.1], [N.sub.2], [N.sub.u], penalty constant [delta] and reference value. Output of the initialization script is calculate matrix of free and forces responses and controller parameters to calculate control value.

In second step was created simulation scheme with Matlab Simulink[R].

[FIGURE 2 OMITTED]

In Fig. 2. is shown Simulink scheme, which was used to simulation. To scheme contains only three blocks. Block "MATLAB Fcn " is used as predictive control, which executes second script (function) for calculate control value in sample time. Block "Discrete Transfer Fcn" is used as controlled system. The last block "Scope" is used to graphic display of control.

The discrete transfer function of system was given in form:

[G.sub.s]([z.sup.-1]) = 0.05075[z.sup.-1] + 0.042[z.sup.-2]/1 - 1.478[z.sup.- 1] + 0.5712[z.sup.-2] [T.sub.0] = 035s (7)

[FIGURE 3 OMITTED]

[FIGURE 4 OMITTED]

[FIGURE 5 OMITTED]

Minimal and maximal horizon of predictive control were determined from step response (see Fig. 3.) and 8. The control horizon Nu was set from 1 to 3. The penalty constant was set [delta] = 1. Reference value was selected in the form of steps.

Parameters of control quality and were calculated subsequently:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The choise of control horizon in case of predictive control had main effect to course of control value and control speed. Especially ther overshoot output value has been influenced. If control horizon value was increased, the control value overshoot was also greater.

5. ACKNOWLEDGEMENTS

This work was supported in part by the Ministry of Education of the Czech Republic under grant MSM 7088352101 and in part by Tomas Bata University in Zlin under grant IGA/55/FAI/10/D.

6. REFERENCES

Camacho, E. F. & Bordons, C. (2004). Model Predictive Control, Springer-Verlag, 978-1-85233-694-3, London.

Mikles, J. & Fikar, M. (2007). Process Modelling, Optimisation and Control, Springer-Verlag, 978-3-540-71969-4, Berlin.

Clarke, D. W.; Mohtadi, C. & Tuffs, P. S. (1987). Generalized predictive control, part I: the basic algorithm. Automatica, 23(2), 137-148, 0005-1098.

Clarke, D. W.; Mohtadi, C. & Tuffs, P. S. (1987). Generalized predictive control, part II: extensions and interpretations. Automatica, 23(2), 149-160, 0005-1098.

Bobal, V.; Kubalcik, M.; Chalupa, P.; Dostal, P. (2009). Adaptive predictive control of nonlinear system with constraint of manipulated variable, Proceedings of the 28th IASTED International Conference on Modelling, Identification and Control, Innsbruck, Austria, pp. 349-354.

Bobal, V. (2008), Adaptive and Predictive control, Tomas Bata University in Zlin, Academia centre, 978-807318-662-3, Zlin (in Czech).
Tab. 1. Control quality

[N.sub.u] [S.sub.y] [S.sub.U]

1 0.0372 0.0051
2 0.0373 0.0028
3 0.0366 0.0028
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Author:Brazdil, Michal; Bobal, Vladimir
Publication:Annals of DAAAM & Proceedings
Article Type:Report
Geographic Code:4EXCZ
Date:Jan 1, 2010
Words:1113
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