# Methods for nonlinear processes control: control structures based on nonlinearity compensators.

1. INTRODUCTION

Various papers and researches target the inverse model control approach; a few of these can be mentioned: (Tao & Kokotovic, 1996), (Pajunen, 1992), (Dumitrache, 2005) etc.

In these researches there have been proposed several types of structures based on inverse model. In accordance with those results, this paper comes up with two very simple and efficient structures presented in Figures 1 and 2. Here, the inverse model is reduced to the process (nonlinear) characteristic.

The first solution (parallel structure) considers the addition of two commands: the first "a feedforward command" generated by the inverse model command generator and the second, generated by a classic, simple algorithm (PID, RST).

The first command, based on the process static characteristic, depends on the set point value and is designed to generate a corresponding value that drives the process's output close to the imposed set point. The second (classic) algorithm generates a command that corrects the difference caused by external disturbances and, accordingly to the set point, by eventual bias error caused by mismatches between calculated inverse process characteristic and the real process.

The second solution (serial structure) has the inverse model command generator between the classic algorithm and the process. The inverse model command generator acts as a nonlinear compensator and depends on the command value. The (classic) algorithm generates a command that, filtered by the nonlinearity compensator, controls the real process.

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The presented solutions propose treating the inverse model mismatches that "disturb" the classic command as some algorithm's model mismatches. This approach imposes designing the classic algorithm with a sufficient robustness reserve.

In Figure 1 and 2, the blocks and variables are as follows: Process--physical system to be controlled; Command calculus--unit that computes the process control law; Classic Alg.--control algorithm (PID, RST); y--output of the process; u--output of the Command calculus block; u alg.--output of the classic algorithm; u i.m.--output of the inverse model block; r--system's set point or reference trajectory; p--disturbances.

Related to classical control loops, both solutions need addressing some supplementary specific aspects: determination of static characteristic of the process, construction of inverse model, robust control law design.

In next sections we will focus on the most important aspects met on designing of the presented structure.

2. CONTROL DESIGN PROCEDURE

For the first structure the specific aspects of the control design procedure are: a) determination of process's (static) characteristic, b) construction of command generator, c) robust control law design of classic algorithm. The second structure imposes following these steps: a) determination of process's characteristic, b) construction of nonlinearity compensator, c) designing the classic algorithm based on "composed process" which contains the nonlinearity compensator serialized with real process. These steps are more or less similar for the two structures. For the (a) and (c) steps is evident; for (b) the command generator and nonlinearity compensator have different functions but the same design and functioning procedure. Essential aspects for these steps will be presented.

2.1 Determination of process characteristic

This operation is based on several experiments of discrete step increasing and decreasing of the command u(k) and measuring the corresponding stabilized process output y(k) (figure 3 (a)). The command u(k) cover all (0 to 100%) possibilities. Because the noises are present, the static characteristics are not identically. The final static characteristic is obtained by meaning of all correspondent positions of these experiments. The graphic between two "mean" points is obtained using extrapolation procedure.

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According to system identification theory, the dispersion of process trajectory can be found using (Ljung & Soderstroom, 1983). This can express a measure of superposing of noise onto process, process's nonlinearity etc. and is very important for the control algorithm robust design.

2.2 Construction of nonlinearity compensator (generator)

This step deals with the process's static characteristic ^transposition" operation. Figure 3 (b) presents this construction. According to this, u(k) is dependent to r(k). This characteristic is stored in a table; thus we can conclude that, for the inverse model based controller, selecting a new set point r(k) will impose finding in this table the corresponding command u(k) that determines a process output y(k) close to the reference value.

2.3 Control law design

The control algorithm's duty is to eliminate the disturbances and differences between the inverse model computed command and the real process behavior. A large variety of control algorithms can be used: PID, RST, fuzzy etc., but the goal is to have a very simple one. For this study we use a RST algorithm. This is designed using pole placement procedure (Landau et al., 1997). Figure 4 present a RST algorithm structure.

The identification is made in a specific process operating point and can use recursive least square algorithm (Landau & Karimi, 1997).

This approach allows the users to verify, and if is necessary, to calibrate the algorithm's robustness. Next expression presents "disturbance-output" sensibility function.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

Based on this value (Landau et al., 1997), some robustness measures like gain margin, modulus margin can be calculated.

For the presented structures, for classic algorithms, it is imposed that the gain margin to be greater or equal to the process static characteristic dispersion: [DELTA] G [less than or equal to] [sigma] ; a controller that has sufficient robustness was designed.

3. EXPERIMENTAL RESULTS

We have evaluated the achieved performances of the proposed control structures using a laboratory experimental installation, presented on Figure 5 (left). Here, the main goal is to control the pressure process, which has a nonlinear characteristic presented in Figure 5 (right). Three real-time software applications, which can be connected with the experimental installation have been designed and implemented using National Instrument LabWindows/CVI and a data acquisition card.

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The tests made during real-time functioning prove the structures stability and performances at set point changing. Figures 6 and 7 present few of these tests.

On these figures the evolution curves are represented using the notations: r(k)--set point; rf(k)--filtered set point; y(k)--process output; u(k)--control structure output (total command). In these tests it can be observed that:

* Both structures are globally stabile and track the set point;

* "Serial" structure tracks the set point very precise but can be sensible at important disturbances ;

* "Parallel" structure doesn't track the set point very precise but can be insensible at important disturbances.

4. CONCLUSIONS

This paper proposes two structures as solution for nonlinear processes control. For each component of these structures, there are presented the design methods. These are based on experimental tests and classic identification and close loop pole placement.

The performances of the classic algorithm command component, is evaluated using robustness criterions.

In experimental results section there are presented the evaluated results obtained using a real time software implementations of proposed control structures. The tests prove good performances of the two structures.

In next researches, we will try to improve these solutions which are limited from the nonlinearity class point of view.

ACKNOWLEDGEMENT

Work supported by IDEI and PARTENERIATE Research Programs of Romanian Research,Development and Integration National Plan II,Grants 1044/2007(Idei), 84/2007 (Parteneriate)

5. REFERENCES

Dumitrache, I., (2005). Numeric control, Politehnica Press, Bucuresti, ISBN 973-8449-72-3.

Landau, I.D. and A. Karimi, (1997). Recursive algorithm for identification in closed loop: a unified approach and evaluation, Automatica, vol. 33, no. 8, pp. 1499-1523.

Landau, I. D., R. Lozano and M. M'Saad, (1997). Adaptive Control, Springer Verlag, London, ISBN 3-540-76187-X.

Ljung L., Soderstroom T., Theory and Practice of Recursive Identification, MIT Press, Cambridge, Massashusetts, 1983, ISBN 10-0-262-12095-X.

Pajunen, G., (1992). Adaptive control of wiener type nonlinear system, Automatica vol. 28, no. 4, pp. 781-785.

Tao, G., and Kokotovic P., (1996). Adaptive control of systems with actuator and sensor nonlinearities, Wiley, N.Y., ISBN-10 047115654X.