# Robust stabilization of interval plants using PI controllers.

1. INTRODUCTION

The mathematical model containing interval parameters is a common tool for description of many industrial processes. This approach helps to compensate the simplifications made during modelling, imprecise knowledge of plant parameters or its variability. However, an interval system itself is only the beginning. The important step consists in designing the controller, which ensures some desired properties of the control loop for the whole interval plant family. And definitely the fundamental requirement of all users is the robust stability of the feedback control system. Despite the existence of many advanced control technologies, the contemporary industrial practice clearly prefers the use of standard PI and PID controllers with fixed parameters and so an easy and effective way of PI/PID tuning is still very topical, especially in case that these algorithms are able to cope with various uncertain conditions.

The main aim of this paper is to present a method of determination of robustly stabilizing PI controllers with fixed parameters for interval systems (Tan & Kaya, 2003) and to demonstrate its capabilities on an example for third order interval plant.

2. COMPUTATION OF STABILIZING PI CONTROLLERS

A possible approach to calculation of stabilizing PI controllers based on plotting the stability boundary locus is proposed in (Tan & Kaya, 2003). The method supposes the classical closed-loop control system with controlled plant:

G(s) = B(s)/A(s) (1)

and PI controller:

C(s) = [k.sub.P] + [k.sub.I]/s = [k.sub.P]s + [k.sub.I]/s (2)

First, one needs to use the substitution s = j[omega] in the plant (1) and subsequently to decompose the numerator and denominator of this transfer function into their even and odd parts:

G(j[omega]) = [B.sub.E] (-[[omega].sup.2]) + j[omega][B.sub.o]/[A.sub.E] (-[[omega].sup.2]) + j[omega][A.sub.o] (- [[omega].sup.2]) (3)

Then, the expression of closed-loop characteristic polynomial and setting the real and imaginary parts to zero lead to the equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Simultaneous solving of these relations and plotting the obtained values into the ([k.sub.P], [k.sub.I]) plane result in the stability boundary locus, which splits the ([k.sub.P], [k.sub.I]) plane up to the stable and unstable regions. The determination of the stabilizing one(s) can be done via a test point within each region. Furthermore, this technique can be embellished with the Nyquist plot based approach from (Soylemez et al., 2003) to avoid potential problems with proper frequency gridding. In this refinement, the frequency axis can be divided into several intervals by the real values of [omega] which fulfill:

Im[G(s)] = 0 (5)

Such intervals are then sufficient for testing.

3. IMPROVEMENT OF THE METHOD FOR INTERVAL SYSTEMS

So far, the area of stabilizing controller coefficients for a given plant with only fixed parameters can be computed. However, the paper (Tan & Kaya, 2003) has improved the stabilization also for interval plants using the simple idea of its combination with the sixteen plant theorem (Barmish et al., 1992), (Barmish, 1994). In compliance with this principle, a first order controller robustly stabilizes an interval plant

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

where [b.sup.-.sub.i], [b.sup.+.sub.i], [a.sup.-.sub.i], [a.sup.+.sub.i] are lower and upper bounds for numerator and denominator parameters, respectively, if and only if it stabilizes its 16 Kharitonov plants, which are defined as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [i.sub.1], [i.sub.2] [member of] {1,2,3,4}; and [B.sub.1](s) to [B.sub.4](s) and [A.sub.1](s) to [A.sub.4] (s) are the Kharitonov polynomials for the numerator and denominator of the interval system (6), respectively.

Remind that the Kharitonov polynomials e.g. for an interval polynomial:

B(s, b) = [m.summation over (i-0)] [b.sup.-.sub.i]; [b.sup.+.sub.i]][s.sup.i] (8)

can be constructed using the upper and lower bounds of interval parameters according to the rule (Kharitonov, 1978):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

The stabilization of an interval plant is grounded in the stabilization of all 16 fixed Kharitonov plants together, and so the final stability region is given by intersection of all partial regions.

4. AN ILLUSTRATIVE EXAMPLE

Suppose that controlled process is described by interval transfer function:

G(s) = [1; 2]/[s.sup.3] + [3;4][s.sup.2] + [5, 6]s + [7; 8] (10)

and the objective is to find all possible robustly stabilizing PI controllers.

First, consider e.g.:

[G.sub.1,1](s) = [B.sub.1](s)/[A.sub.1](s) = 1/[s.sup.3] + 4[s.sup.2] + 5s + 7 (11)

as the first of 16 Kharitonov plants. The equation (4) here takes the concrete form:

[k.sub.P] = 4[[omega].sup.2] - 7 [k.sub.I] = -[[omega].sup.4] + 5[[omega].sup.2] (12)

Using of (5) and consequent stability test for two obtained intervals lead to the range of the frequency [omega] [member of] (0; 2.236), which is necessary for computing/plotting the stability boundary locus. The analogical procedure has been done generally for all 16 Kharitonov plants. However, in this specific case, the locus of only 8 systems is enough to investigate, because the nominator of (10) takes only two extreme values and the construction of Kharitonov polynomials would be redundant here.

The fig. 1 provides the graphical representation of the stability boundary locus for 8 Kharitonov plants, while the fig. 2 brings closer look to the intersection, which constitutes final stability region for the interval plant (10). An arbitrary pair of ([k.sub.P], [k.sub.I]) from inside of this stability region would entail robustly stable control system.

[FIGURE 1 OMITTED]

[FIGURE 2 OMITTED]

5. CONCLUSION

The contribution has presented a possible robust stabilization technique for interval systems using PI controllers with fixed parameters. The third order interval plant has been successfully stabilized in the illustrative example.

6. ACKNOWLEDGEMENT

The work was supported by the Ministry of Education, Youth and Sports of the Czech Republic under Research Plan No. MSM 7088352102. This aid is gratefully acknowledged.

7. REFERENCES

Barmish, B. R. (1994). New Tools for Robustness of Linear Systems, Macmillan, New York, USA.

Barmish, B. R.; Hollot, C. V.; Kraus, F. J. & Tempo, R. (1992). Extreme point results for robust stabilization of interval plants with first order compensators. IEEE Transactions on Automatic Control, Vol. AC-37, pp. 707-714.

Kharitonov, V. L. (1978). Asymptotic stability of an equilibrium position of a family of systems of linear differential equations. Differentsial'nye Uravneniya, Vol. 14, pp. 2086-2088.

Soylemez, M. T.; Munro, N. & Baki, H. (2003). Fast calculation of stabilizing PID controllers. Automatica, Vol. 39, No. 1, pp. 121-126.

Tan, N. & Kaya, I. (2003). Computation of stabilizing PI controllers for interval systems, In: Proceedings of the 11th Mediterranean Conference on Control and Automation, Rhodes, Greece.
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