Inverted pendulum: construction, model and controllability.

Ciontos, Ovidiu ; Dolga, Valer

1. INTRODUCTION

The inverted pendulum is a classic example of nonlinear system, unstable, with one input signal and multiple output signals.(Dolga, 2007) Systems with inverted pendulum are a good testing and demonstration of techniques for automatic control. Stability of inverse pendulum is a base model to control the walked robots, rocket thrusters, balancing heavy loads with cranes, etc (Huang & Huang, 2000).

Many researchers have focused to control an inverted pendulum in unstable equilibrium. Balancing of inverted pendulum was study in modern control theories such as fuzzy control, nonlinear control and neural networks. Control and simulation is done with programs like: Matlab/Simulink, WinCon, Real Time Workshop, etc. (Magana & Holzapfel, 1998)

Controllability and observability of systems with inverted pendulum has been studied only in practical examples (Chen & Chen, 1998),(Kajiwara et al.,1999),(Ignat & Rusu, 2008). This paper presents a synthesis of the nonlinear aspects of the system and a generalization for the controllability of these systems.

Studies and experimental researces aimed at obtaining an model who allowing an relevant analysis of the system, parameters identification and statement of principles for optimal synthesis of mechatronic systems. The goals are also the onces fot the phd thesis. Future research is desired to control an inverted pendulum with a pneumatic bench.

2. THE INVERTED PENDULUM

Classical problem of inverted pendulum control is interesting because it can be solved using a wide range of solutions principled and constructive alternatives.

The system consists mainly of:

* basic support in a possible move of translation or rotation.

* pendulum articulated on this support.

In Fig. 1 authors presented the proposal for taxonomy of inverted pendulum.

The constructive variants differ from the chosen solutions for the base support and concretization module of the translation movement. For the variant from Fig. 2f the 2D inverted pendulum is articulated at the base support through a universal joint. There are also worth mentioning the solutions for the translation support offered by the belt driver unit, screw driver unit and linear sliding unit bearing.

[FIGURE 1 OMITTED]

Aspects of principled solutions and constructive variants of the inverted pendulum are presented in Fig. 2.

[FIGURE 2 OMITTED]

[FIGURE 3 OMITTED]

3. MATHEMATICAL MODEL OF INVERTED PENDULUM SYSTEM, CONTROLLABILITY

Considering the inverted pendulum system in his integrity, the mathematic model of system is composed by the actuator model, base support model(movement transmission model, movement guide module model), pendulum model. This model can be described as a generalized:

dx/dt = F[x(t), u(t), t]

y(t) = G[x(t), u(t), t] (1)

where: x(t) is the status vector of the system, u (t) is the input vector of the system, and y (t) is the output vector. In a compact mod mathematical model of the system can be described by differential equation of state and output equation:

dx/dt = A x + B x u

y = C x + D x u (2)

where matrix A is the state matrix of the system, B is the matrix order, C is the output matrix, and D is the reaction matrix.

The system presented is strongly nonlinear because of:

* the non-linearity of the friction module present in the actuator, the guidance module, the joint of the pendulum.

* the non-linearity of the actuator (elasticity, looseness).

* the non-linearity of mathematical module of the pendulum (transformations through the functions "sin", "cos").

For the nonlinear system can be established one balance point, [[x.sub.0], [u.sub.0], [y.sub.0]], in motion. This point fulfills the equation system:

0 = F[[x.sub.0], [u.sub.0]]

[y.sub.0] = g[[x.sub.0], [u.sub.0] (3)

The linearity of the admitted system is done around this functioning point. In the new linearised form the ecuations (2) are:

d[DELTA]x/dt = A x [DELTA]X + b X [DELTA]u

[DELTA]y = C x [DELTA]x + D x [DELTA]u (4)

where:

* [DELTA]x = x(t)--[x.sub.0], [DELTA]u = u(t)--[u.sub.0], [DELTA]y = y(t)--[y.sub.0] (5)

* [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Taking into account the state variables x, [??], [phi], [??] and as output for the base module in translation x and as for pendulum [phi] the linearised model for the variant from Fig. 2a has the matrices A and B have this form:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

where [m.sub.i] is the mass of the translation module respectively the pendulum one, I is the inertia moment of this and [k.sub.v] is the coefficient viscous friction.

Based on the previous matrices, one can define the controllability matrix:

[[GAMMA].sub.C][A, B] = [B AB ... [A.sup.n-1]B] (8)

4. CONCLUSION

Controllability has an important function in regulating many problems, such as stabilizing a system with feedback or optimal control

If all the sizes of the model state of a system are controllable, then the system is fully controllable. The model is controllable if and only if the controllability matrix of the system, defined by the relation (8), has full row rank.

Calculations showed that regardless of the parameters of equation (7), the controllability matrix has full row rank, so the inverted pendulum is fully controllable

5. REFERENCES

Chen, C.S. & Chen, W.L. (1998). Robust Adaptive Sliding-Mode Control Using Fuzzy Modeling for an Inverted-Pendulum System. IEEE Transactions on Industry Applications, Vol. 45, No. 2, (April 1998) page numbers (297-306), ISSN: 0278-0046

Dolga, V. (2007). Mechatronics Design, Editura Politehnica, ISBN 978-973-625-573-1, Timisoara

Huang, S.J. & Huang, C.L. (2000).Control of an Inverted Pendulum Using Grey Prediction Model. IEEE Transactions on Industry Applications, Vol.36, No.2, (March/April 2000) page numbers (452-458), ISSN: 0093-9994

Ignat, I. & Rusu, M. (2008). Design and build a stand for the study and control of the rotational inverted pendulum Available from : www.fsc.ugal.ro/resources/evenimente/ 2008_aii_premiul_III_Ignat_Ionel.pdfAccessed: 2009-0611

Kajiwara, H.; Apkarian, P. & Gahinet, P. (1999).LPV Techniques for Control of an Inverted Pendulum, Control Systems Magazine, IEEE, Vol. 19, No. 1 (February 1999) page numbers (44-54), ISSN: 0272-1708

Magana, M.E. & Holzapfel, F. (1998). Fuzzy-Logic Control of an Inverted Pendulum with Vision Feedback. IEEE Transactions on Industry Applications, Vol. 41, No. 2, (May 1998) page numbers (165-170), ISSN: 0018-9359