Designing, implementing and analysing optimal controllers on a non-linear reaction wheel pendulum

This paper discusses a reaction wheel control system. A reaction wheel pendulum is good example of a non-linear and underactuated system, which attracts control system researchers to develop many control algorithms. The reaction wheel plant is usually used for studying advanced control system courses. In this paper, a mathematical model of the state space is discussing. A proposed LQR control algorithm is explained. Simulation and real time experiments have seen carried out to verify the performance of the proposed closed loop system. The LQR optimisation algorithm was able to find the optimum feedback gains. The simulation and real time experiments show that the reaction wheel pendulum could stabilize the pendulum at upright position.


Introduction
Underactuated mechanical systems are well known of systems that present interesting characteristics for test linear, nonlinear and intelligent control algorithms, since this kind of systems represent an interesting challenge from the viewpoint of control theory and engineering [1]. Pendulums are present everywhere around us, translated in different forms. To give a definition, a pendulum is simply an object swinging freely relative to its equilibrium position [2]. A Pendulum system has been a popular demonstration platform for nonlinear and underactuated control at least fifty years [3]. There are some control experiments performed in different pendulum systems or balancing system. It is typical to use one control law to swing up the pendulum ad another to balance it around the upright position [4]. In recent time optimal control provides the best possible solution to process control has been presented in [5].
One of pendulums is reaction wheel pendulum. The reaction wheel is a spinning mass which provides a reaction torque coming from the rotational acceleration [6]. The Reaction Wheel Pendulum is perhaps the simplest of the various pendulum systems in terms of its dynamic properties, consequently, its controllability properties. The Reaction Wheel Pendulum is one of newest modification form of pendulum. The Reaction wheel pendulum is a system contains from pendulum and rotating disk. The Reaction Wheel Pendulum is a simple pendulum with a rotating wheel, or bob, at the end. The level of background knowledge assumed is that of a first course in control, together with some rudimentary knowledge of dynamics of physical systems [7]. The Reaction Wheel Pendulum exhibits several properties, such as underactuation and nonlinear, that make it an attractive and useful system for research and advanced education [8]. The Reaction Wheel Pendulum is suited for education for university students.

Lagrange's equation
Represented the motion equation of a complex system dynamic is using Lagrange's equation. The kinetic energy is kinetic energy from the movement relative to the center of mass and the kinetic rotational energy about the center of mass. So total kinetic energy is The potential energy of the reaction wheel pendulum is obtained as follows The Lagrange function is expressed by the kinetic energy and the potential energy . Where the kinetic energy function in terms of the generalised coordinate and its derivative ̇.   By substituting the parameters values in Table 1, the state -space expression of the system is changed into: [∅∅̈̈] = [ (2.13)

LQR method
Linear Quadratic Regulator Controller is based on full state -space feedback control principle ( Figure  2) [9]. The optimal linear Quadratic control method aims to obtain a controlling signal li u(t) that will move a linear system state from the initial condition x(t_o) to the final condition x(t) that will minimize the cost function, expressed as.
Matrix selection and ℛ doing by trial and error. Where matrix is symmetrical matrix, positive semidefinite ( ≥ 0), while matrix ℛ is positive and real (ℛ > 0). The relative importance of the error and the energy cost are determined by and . Therefore, The matrix and signify the trade-off between performance and the control effort respectively.
From figure 3, the denominator can be found using formula as Where I is an identity matrix. So, the stability and transient response characteristics of the closed loop system are determined by all eigenvalues of ( − ) . The design is to select the feedback gain K such that eigenvalues of ( − ) ( − ) have negative real parts [10].

Result and analysis
Simulated testing has been done using MATLAB. Figure 3 and Figure 4 show the system response.  Figure 3. Graph response with control with gain feedback (K).
From figure 3 the curve in red represents the wheel's angle in radians, and the curve in blue represents pendulum angle in radian. As you can see, this plot is not satisfactory. Wheel pendulum and pendulum angle's overshoot appear fine, but their settling times need improvement. So, Need to do changes to the value .  Using the value of parameter system from table 3, obtained   Figure 4 show the graph with higher value than before. With changing the value of can make the system becomes more stable.

Conclusion
Reaction wheel Pendulum is a non-linear system and underactuation system. This system is unstable and can be stable when modelled by using mathematical equations. From this experiments we get the value of gain feedback [-534.5587 -111.3672 -10.000 -4.4596] to stabilize the reaction wheel pendulum in simulation. From the picture of graphic is known that system can stabilize and can be at the upright position.