System Modeling Approaches
The double inverted pendulum is an age old complex system of sorts normally used as a yardstick to ascertain the efficacy of any devised method of control due to its structural simplicity. Generally, in stably controlling a double inverted pendulum, there are six input items required for the two pendulums in all the angular controls and a suitable cart control position. Studies show that there several methods that can be employed in the derivation of equations aimed at expressing the principle of double inverting. As an example, Zhong and Rock (2001) demonstrate the possibility of applying Newtonian approaches in the description of this principle. In addition, they propose the use of the Euler-Lagrange approach. This approach entails the utilization of the Euler-Lagrange equation together with the Lagrangian. Various approaches have been put forward, but in most cases the Lagrange approach is employed due to its simplicity. Unique specifications are usually put in place in the process of isolating the most suitable system requirements. Mechanical damping characteristics need to be identified and these are in turn used in choosing the most suitable approach when dealing with linear systems (Rubi and Avello 2002). The specifications of the invariable DC motors should also be stated according to Rubi and Avello (2002). The determination of the amount of noise produced during the measurement process is a key factor that has been suggested in the construction of actual systems (Katupitiya and Luong 2001).
Inverted Pendulum Designs
According to Dorf and Bishop (1998), the most basic design of an inverted pendulum comprises of a pendulum link that is joined to a powered motor shaft. Even though the system works, it cannot be of great use in the real world. However, for this system to be applicable, it should have at least two angles of freedom. The system can be modified in such a way as to increase the degree of freedom (Dorf and Bishop 1998).
Depending on the configuration, the pendulum is joined to the cart through various methods (Franklin 2002). Ideally, researchers claim that there is unlimited number of links can be used with promising results. The double inverted pendulum whether linear or rotary, entails the use of two link systems and each of these links has only a single degree of freedom (Rubi and Avello 2002).
Control Design
The design of the control system employs variation of the configuration of the links. Furthermore, there have been suggestions of the utilization of a linear quadratic to increase the performance especially in the region of the feedback regulator (Zhong and Rock 2001). In addition, it has been suggested that the use of the sliding approach provides augmented stabilization of equilibrium in the vertical configuration (Utkin, et al. 1999). A more accurate model of the friction should include static and coulomb friction and was therefore considered in two models that incorporated static and coulomb friction thus creating smoothness from the cart being stationery than when it was moving (Kaelbling, et al. 240). Another approach was the use of robust control techniques. These robust control techniques control have been applied to both single and double inverted pendulum systems (Crites and Barto 407). The incorporated controllers system provided a system where the amplitude related to the unmodelled friction was greatly minimized (Kaelbling, et al. 242). Other studies have demonstrated the compatibility of the inverted pendulum with several systems. An example is the study that used the neural network technique and a single bar reinforced technique. Both techniques were not successful (Crites and Barto 407).
Works Cited
Crites, Robert, and Barto, Andrew. “An actor/critic algorithm that is equivalent to Q-learning,” Advances in Neural Information Processing Systems 7.1 (1994): 401-408.
Dorf, Richard, and Bishop, Robert. Modern Control Systems. Los Angeles: Addison Wesley Longman, 1998.
Franklin, Gene. Feedback Control of Dynamic Systems. New Jersey: Prentice Hall, 2002.
Kaelbling, Leslie, et al. “Reinforcement learning: A survey.” Journal of Artificial Intelligence Research, 4.1 (1996): 237-285.
Katupitiya, Van Jayantha and Luong, Daniel. An adaptive kalman filter with quadrature encoder quantisation compensation. University of New South Wales: Sydney, 2001.
Rubi, Jose, and Avello, Ángel. “Swing up control problem for a self erecting double inverted pendulum.” IEE Proceedings of Control Theory Application 149.2 (2002): 23-50.
Utkin, Vadim, Gulder, Juergen, and Shi, Jingxin. Sliding mode control in electromechanical systems. Pudstow: T.J. International Ltd, 1999.
Zhong, Wei, and Rock, Helmut. “Energy and passivity based control of the double inverted pendulum on cart.” IEEE Conference on Control Applications, 5.1(2001): 896-901.