INPUT-OUTPUT FEEDBACK LINEARIZATION ASSOCIATES WITH LINEAR QUADRATIC REGULATOR FOR STABILIZATION CONTROL OF FURUTA PENDULUM SYSTEM

: Manuscript provides a key technology, namely Input-Output Feedback Linearization Associates with Linear Quadratic Regulator (for short, IOFLALQR). The objective of this research is to study the possibility of integrating two control strategies, which includes input-output feedback linearization technique (for short, IOFL) and linear quadratic regulator controller (for short, LQR), for stabilization control of Furuta pendulum system. Furuta pendulum system belongs to the group of under-actuated robot systems. In this work, structure of IOFLALQR, control implementation, comparison of IOFLALQR and conventional LQR are adequately studied and discussed. Simulation is completed in MATLAB/Simulink environment and experiment is done on real-time experimental setup. Numerical simulation and experimental results show that the IOFLALQR are implemented on Furuta pendulum successfully. Besides, results have been drawn for demonstrating IOFLALQR better than another classical method.


Introduction
Main purpose of combination two or three control strategies is to validate ability of integrating those techniques together for stabilization control, swing-up control and more.Many researches have been completed and shown to scientific community related to topic of combining control approaches.Diversity of combination of controllers have been presented such as a combination of sliding mode control (SMC) and LQR for stabilization control of Quadcopter [1], combining SMC with variable weights and LQR for trajectory tracking control problem for dual-motor autonomous steering system [2], sliding mode -disturbance observer (SMC-DO) combines with LQR technique for controlling flexible manipulators robot [3], controlling 3D overhead crane systems by using PID-SMC [4], adaptive back-stepping sliding mode [5], optimum fuzzy combination of decoupled SMC (DSMC) [6], a feed-forward controller combines with feedback controller for tracking control problem [7], adaptive radical basis function neural networks associates with proportional derivative-SMC method [8], adaptive fuzzy logic back-stepping methodology [9], combination of state feedback controller with RBF [10], SMC integrates with partial feedback linearization for a spatial ballbot [11].In addition, combination of control strategies for swing-up problem have been studied.For instance, research of combining two strategies consists of deep reinforcement learning and local control for swinging up acrobot [12], feedback linearizationenergy control method combination for swing-up control of rotary inverted pendulum (RIP) [13], an experiment of combination of on-off and SMC methods for swinging up a pendulum with two reaction wheels [14], Qlearning and PID controllers combination for swinging up a nonlinear double inverted pendulum [15] and more.In this work, we develop an IOFLALQR from IOFL method.The new suggested control algorithm is qualified to stabilize a fourth-order under-actuated nonlinear Furuta pendulum system.Furuta pendulum system, named after Japanese Professor Katsuhisa Furuta, was invented at Tokyo Institute of Technology.The first study was completed on this system in 1992 related to an application of pseudo-state feedback for swinging up an inverted pendulum [16].Currently, Furuta pendulum or RIP, can find easily in laboratories related to control engineering.Furuta pendulum is single-input multi-output system (SIMO).In this paper, we validate the proposed control scheme on this Furuta pendulum system, which is available at Control System Laboratory, HCMUTE.Many researches have been done on this system such that back-stepping control scheme [17], LQR technique [18], SMC [19], IOFL [20], PID-Neural controller [21].
Feedback linearization consists of two control strategies: IOFL and exact state-space linearization.In this paper, we focus on input-output linearization approach.This method transforms a certain class of nonlinear systems into linear systems by a proper coordinate change and a linearizing state feedback [22].Detail of this control scheme can be found in [23].LQR is able to overcome big disturbance is going on stability the system without reducing working performance and can overcome disturbances that occurred previously [24].
Main contribution of this paper are proposed new technology, namely IOFLALQR.This control strategy is conducted on both simulation and experimental setup.After that, we compare the state responses of output system between IOFLALQR and LQR technique.

Mathematical Model
Structure of RIP is shown in Fig. 1 Fig. 1.RIP Parameters of system are listed in Table 1.These parameters are measured from real model in Fig. 2. According to [25], equations of of RIP are presented: where, ( ) ( ) sin ( ) cos () cos where 1 The state-space equations of RIP are given below.Consider 12 x  = as output of system.We obtain where ( ) ( ) Linearized system operating around the equilibrium point is described as follows:  From [26], we analyze the stability of system.Let 0 x = be an equilibrium point of nonlinear system () where :D n f → is continuously differentiable and D is a neighborhood of the origin.Let 0 () 1.The origin is asymptotically stable if Re 0 i   for all eigenvalues of A.
2. The origin is unstable if Re 0 i   for one or more of the eigenvalues of A.
RIP has an equilibrium point   With results in (7), there is one eigenvalue in the open right-half plane.Hence, the system is unstable at equilibrium point.

Experimental Setup
The real-time experimental setup is similar with the experimental setup [20].

Principles of IOFL and application of IOFL for
Furuta pendulum are mentioned in [20].The calculation of control law bases on the state-space equation ( 3), which follows closely the designed controller in [20].In this manuscript, we re-use the control law that mentioned in [20].Schematic diagram of IOFL is shown in Fig. 3.  ; K is a positive matrix that need to be selected with n is a number of degrees of system; z is states vector: In addition, by using genetic algorithm (GA), the controlled parameters

LQR
LQR controller is widely used in control engineering for stabilization control under-actuated system like inverted pendulum, pendubot, etc.General control law of LQR technique is where,

 
129.6261 23.8921 5 6.0310 is calculated from weighing matrixes Q, R from GA searching, linear matrices A, B from (4) and system parameters in Table 1.These matrices are listed as below

IOFLALQR
The advantage of LQR is simple structure and wellstabilizing RIP around working point.However, the working space is just being around the working point.Differently, nonlinear control, such as IOFL method, has wide range of working due to flexible structure of controller and stability proved from Lyapunov criteria.Disadvantage of IOFL method is the difficulty in selecting function h(x) in (8).With the selection of h(x)=x1, only the angle of pendulum is guaranteed.The motion of arm is not kept closed to zero value.Following to [20], the arm cannot be kept closed to zero point.It moves around the working point, about 20 degrees with sine form.Then, we propose a hybrid controller which can combine LQR and IOFL to stabilize RIP at working point as in Fig. 4.

Fig. 4. Hybrid controller in controlling RIP
The introduced RIP controller is based on a combination of input-output feedback linearization (IOFL) and linear quadratic regulator (LQR).Simulink schematic of representation of IOFLALQR method is drawn in Fig. 5.

Numerical Simulation Results
In this subsection, simulation results are shown in Fig. 6 and Fig. 7. Content of Fig. 6 is performance of state responses of output system by implementing IOFLALQR.In Fig. 6, from top to bottom, graphs are organized as follows: state response of angular position of pendulum 2 Following first graph of Fig. 7, angle of pendulum is back to equilibrium point after 2 seconds, maximum overshot range of this state is [-0.03;0.1](rad).In third graph, angle of arm stabilizes at "0" (rad) after 4 seconds and maximum overshoot range of this state is [-0.3;0.1](rad).Angular velocity of pendulum and arm are depicted in 2 nd and 4 th graph of this figure.We can observe that system under IOFLALQR can stabilize at equilibrium point.Moreover, control input of RIP is described 5 th graph.
Moreover, from Fig. 7, system with LQR controller has more significant overshoots in angular displacements of pendulum and arm.From simulation results, it can be observed that system with IOFLALQR has better performance, in terms of less overshoot, faster convergence in pendulum angle, and arm angle.In addition, control input of two controllers are also compared in last graph of Fig. 7.

Experimental Results
The main purpose of this subsection is to provide performance of RIP via experiment.Experiment results are drawn in Fig. 8. Contents of Fig. 8 are performance of state responses of output system by implementing IOFLALQR and conventional LQR techniques.In Fig. 8, from the top to bottom, graphs are organized as follows: state response of angular position of pendulum 2  (rad), state response of angular velocity of pendulum 2  (rad/s), state response of angular position of pendulum 1  (rad), state response of angular velocity of pendulum 1  (rad/s), control input (V).State responses of system under IOFLALQR are described in blue and state responses of system under LQR are described in orange.Following the first and third graph, pendulum angle has a minor oscillation around equilibrium point and better performance than another, while the angle of arm of system with IOFLALQR has better performance than conventional LQR.The angular velocity of arm and pendulum of system with IOFLALQR and conventional LQR are also captured in the 2 nd and 4 th graph of this figure.The comparison of voltage input is also provided in the last graph of Fig. 8.

Conclusion
The purpose of this paper is to qualify the proposed stabilization control of nonlinear RIP system, namely IOFLALQR.The stabilizing control laws of RIP were designed, realized, and experimentally tested.IOFLALQR was designed based on combination of input-output feedback linearization (IOFL) and linear quadratic regulator (LQR).The results showed that the proposed control law guarantees the closed-loop system to be asymptotically stable.The experimental setup was built, and the controllers were realized.The designed control system demonstrated to be effective in simulation and experimental results.In the simulation studies, IOFLALQR control scheme yields a smaller pendulum angle deviation, a smaller arm angle deviation, and a smaller angular velocity of arm and pendulum deviation than conventional LQR.In the experimental work, IOFLALQR was applied successfully on physical experimental RIP.The future work can be realized such as validation of IOFLALQR on other under-actuated system like inverted pendulum, etc.
= , it means that (A, B) is controllable.The eigenvalues of A are

Fig. 3 .
Fig. 3. IOFL control scheme Control signal of this method is

Fig. 6 .
Fig. 6.State responses of output system under IOFLALQR method

Fig. 7 .
Fig. 7. Comparison of state responses of output system under IOFLALQR and LQR techniques

Fig. 8 .
Fig. 8.Comparison of state responses of output system with IOFLALQR and LQR techniques

Table 1 . System parameters Param eter Value Unit Description m
Feedback Linearization Associates with Linear Quadratic Regulator for Stabilization Control of Furuta Pendulum System ), state response of angular velocity State responses of system under IOFLALQR are described in blue and state responses of system under LQR are described in orange.Original points are set up for this simulation as follows: