An energy saving method of stable control of inverted pendulum system when affected by external interference using auxiliary pendulum

. This research aims to develop a method to reduce energy consumption when controlling an inverted pendulum system that is affected by external interference. In this paper, the authors use the quasi time-optimal control law and add on an inverted pendulum an auxiliary pendulum to absorb the energy of the external interference effects, to reduce the cost of controlling the energy stable inverted pendulum while ensuring system quality. The quality of the method is demonstrated through simulation results. The effectiveness of this method is shown by comparison with the method of no damping.


Introduction
Inverted pendulum systems have many industrial and defense applications, for example: two-wheel self-balancing (seg-way) vehicles, missiles, self-propelled ammunition, smart robots, rigs and crane systems [10]. This is a multi-input and multiple-output mechanical system (MIMO) with high nonlinearity, instability [11]. On real objects, the cost of energy controlling the system is also an urgent issue to be researched. The works [1][2][3][4] use time optimal control laws for inverted pendulum systems, but with high energy costs, therefore are difficult fit in embedded systems. In the works [5][6][7][8][9], the authors have not introduced the energy saving problem, only interested the quality of the control system. In this paper, the authors presents a quasi time optimal control method and attaches on an inverted pendulum an auxiliary pendulum with elastic coefficient and dissipation coefficient to reduce system energy costs and control quality is not affected.
The inverted pendulum model is mounted on an inverted pendulum with an auxiliary pendulum illustrated in Figure 1. In which the auxiliary pendulum has a elastic coefficient k and dissipation coefficient c. Assume that the friction coefficient and moment of inertia of the bar are negligible. The model of linearization math at the point (0,0,0) of the inverted pendulum system has an additional damping device of the form: In which: x -cart position (m); ẋ -cart speed (m/s); ẍ -cart acceleration (m/s 2 ); 1 2 ,    deflection angle (rad); 1 2 ,     -angular velocity (rad/s); 1 2 ,     -angular acceleration (rad/s 2 );

The synthesizing of quasi time-optimal control law for the pendulum systems
The nearly optimal method by quick acts has been applied in some studies [1,2,5]. This method has many advantages such as fast duration effect, asymptotic stability, and high stability. In the paper [10], development of control rules on embedded systems with the one -free -step object is shown. Using a nonlinear model for the two -free -step inverted pendulum systems is slightly complicated. Therefore, the authors use the linear system (1) for embedded systems.
Changing state variables as follows: It is easy to verify that Jacobian matrix of transformation (4) is not degraded: Where: Equations by Equations (5) is in a controllable Jordan form [8]. Virtual system (5), which has asymptotic stablily and quasi -time optimum, is used to synthesize the control law F with z 1 = y 1 ; F is found to introduce the system (5) into the virtual system (6) [8]. The obtained control law is a differentiable function of state variables, of which the formula is very long and therefore not given here.

Simulation results of an inverted pendulum system with auxiliary pendulum
The selected parameters for the simulation model are as The graph of Figure 3 shows the simulation results of the force acting on the inverted pendulum system when there is an external interference effect. It is easy to recognize that the cost of force acting on the inverted pendulum system is often much greater than the force cost acting on the inverted pendulum system with an attached auxiliary pendulum. The total cost of force acting on the system during the simulation time of 2 s is in sequence: 4769.6 (N) and 21.39 (N). This shows the efficiency of the proposed method of saving the cost of controlling the inverted pendulum system.

Conclusion
The paper presented a method to reduce energy costs for the inverted pendulum system using the quasi time-optimal optimal control law. It is the use of auxiliary pendulum mounted on the main inverted pendulum. Although there is a reduction in system dynamics, the pendulum will absorb the interference effect, increasing the stability of the system. The simulation results have shown that the quality of stable control of inverted pendulum system is well met. Specially reducing the system energy cost when stabilizing the inverted pendulum has the effect of external interfenrence.