# Control of unstable systems a polynomial approach.

1. INTRODUCTION

Many dynamic systems in industry or transport have an unstable behavior. A classical control theory with traditional PID controllers often hardly overcomes the stabilization of unstable systems; see e.g. Astro'm and Hagglund (1995). However, the appropriate of the feedback controller can change this unstability. Naturally, also an open-loop stable system can become unstable after feedback control, which is undesirable. Therefore, stability is a fundamental requirement for any controlled feedback system.

An effective and attractive tool how to change the stability properties can be found in algebraic methods, see Kucera (1993), Prokop and Corriou (1997). The general problem for unstable systems was solved in Prokop et al. (2001). Diophantine equations in the polynomial ring are simple and proper tool how to design suitable controllers. The fundamental task is to express a feedback characteristic equation with a stable right hand side. The system design approach utilizing the Diophantine equation is not a state-variable system design technique. It is a "purely algebraic" approach that deals with transfer functions of the controlled plant and controller. The main reason for considering Diophantine equation in this paper is that it allows to address unpleasant but very realistic situations when the controlled plant has right-hand-side zeros.

2. STABILIZATION

The easiest way to check the stability of a linear continuous system is to check the pole locations in the complex plane. If there is a pole with a positive real part, the system is said to be unstable and this pole is referred to as an unstable mode of the system. In other words, if there is one or more poles on the right half plane, the system is unstable. The system is stable if all poles have negative real parts, that is, all poles lie in the left half plane. The imaginary axis then represents a stability border. The systems with poles on imaginary axis are either oscillating or integrating ones, both unstable.

[FIGURE 1 OMITTED]

Suppose the feedback system depicted in Fig.1 where G(s) is a controlled (unstable) system and C(s) is a controller. Both are expressed as a ratio of polynomials:

C(s) = q(s) / p(s) (1)

G(s) = b(s) / a(s) (2)

and a reference signal is also defined as a ratio:

W(s) = g(s) / f(s) (3)

The basic Diophantine equation for the stabilization problem in the ring of polynomials can be read as

a(s) x f(s) x p(s) x b(s) x q(s) = m(s), (4)

Where m is a stable polynomial with the appropriate degree,

see Prokop and Corriou (1997), Prokop et al. (2001). The control law is then given by

f (s) x p(s)U (s) = q(s)E(s) (5)

3. CONTROL DESIGN

According to mentioned methodology, first and second order systems were assumed as unstable ones:

[G.sub.1](s) [b.sub.0] / s+a (6)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Then the controllers take the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (9)

for the first and second orders, respectively. The right hand side polynomials are then

[m.sub.1](s) = [(s + [m.sub.0]).sup.2] (10)

[m.sub.2](s) = [(s + [m.sub.0]).sup.4] (11)

Where [m.sub.0]>0 is a multiple pole (positivity follows from stability) and the value of [m.sub.0] can be used as a tuning knob for influencing of response properties.

4. EXAMPLES

1) Let transfer function will be [g.sub.1] (s) = 1 / s - 0,5

Diophantine equation then takes the following form

(s - 0,5) x s x [p.sub.0] + 1 x ([q.sub.0] + [q.sub.1]s) = [(s + [m.sub.0]).sup.2]

and the comparison of appropriate coefficients gives a set of linear algebraic equations:

[S.sup.2]: [p.sub.0] = 1

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

[S.sup.0]: [q.sub.0] = [m.sup.2.sub.0]

The final controller for [m.sub.0]=2 is

[C.sub.1](s) = q(s) = p(s)f(s) = 4+4,5s / s

2) Let transfer function will be [G.sub.2] = 1 / [s.sup.2](s) -s - 2

Diophantine equation then takes the following form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the comparison of appropriate coefficients gives a set of linear algebraic equations:

[s.sup.4]: [p.sub.1] = 1

[s.sup.3]: [p.sub.0] - [p.sub.1] = 4[m.sub.0]

[s.sup.2]: - [p.sub.0] - 2[p.sub.i] + [q.sub.2] = 6[m.sup.2.sub.0]

[s.sup.1]: - 2[p.sub.0] + [q.sub.1] = 4[m.sup.3.sub.0]

[s.sup.0]: [q.sub.0] = [m.sup.4.sub.0]

The final controller for [m.sub.0]=0.5 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

The final controller for m0=2 is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A Matlab Simulink was created for design and simulation of controllers, see Dingyii Xue, YangQuanChen, Derek P. Atherton (2007). First and second order systems are considered as nominal plants. The Simulink scheme of feedback system is shown in Fig. 2

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

6. CONCLUSION

A design method based on converting an originally unstable system into a stable feedback system was developed for unstable systems generally. Resulting control laws are of PID types. The proposed method enables to tune and influence of response properties and the control behavior by a single parameter m0 (tuning knob). A Matlab Simulink was developed for controllers design and simulation.

7. REFERENCES

Astrom, K.J. and T. Hagglund (1995). PID Controllers: Theory, Design and Tuning. Instrument Society of America, USA

Dingyu Xue, YangQuanChen, Derek P. Atherton (2007). Linear feedback control: analysis and design with MATLAB. ISBN 978-0-898716-38-2 (alk. paper)

Kucera, V. (1993). Diophantine equations in control--A survey, Automatica, Vol. 29, No.6, pp. 1361-1375

Prokop, R. and J.P. Corriou (1997). Design and analysis of simple robust controllers, Int. J. Control, Vol. 66, No. 6, pp. 905-921

Prokop, R., P. Hustak and Z. Prokopova (2001). Simultaneous tracking and disturbance rekjection for unstable systems. In: Prepr. of 13th Conf.on Process Control 01, Strbske Pleso