Real-Time Swing-up and Stabilization Control of a Cart-Pendulum System with Constrained Cart Movement

Ashish Singla; Gurminder Singh

doi:10.1515/ijnsns-2017-0040

Enjoy 40% off

academic books on De Gruyter Brill *

Article

Real-Time Swing-up and Stabilization Control of a Cart-Pendulum System with Constrained Cart Movement

Ashish Singla and Gurminder Singh

Published/Copyright: September 14, 2017

Published by

Become an author with De Gruyter Brill

Author Information

From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 18 Issue 6

Abstract

The cart-pendulum system is a typical benchmark problem in the control field. It is a fully underactuated system having one control input for two degrees-of-freedom (DOFs) system. It has highly nonlinear structure, which can be used to validate different nonlinear and linear controllers and has wide range of real-time (realistic) applications like rockets propeller, tank missile launcher, self-balancing robot, stabilization of ships, design of earthquake resistant buildings, etc. In this work, modeling, simulation and real-time control of a cart-pendulum system is performed. The mathematical model of the system is developed using Euler-Lagrange approach. In order to achieve a more realistic model, the actuator dynamics is considered in the mathematical model. The main aim of this work is to investigate the performance of two different control strategies- first to swing-up the pendulum to near unstable equilibrium region and second to stabilize the pendulum at unstable equilibrium point. The swing-up problem is addressed by using energy controller in which cart is accelerated by providing a force to the cart with a AC servo motor with the help of timing pulley arrangement. The initial velocity of the cart is taken into account to confirm swing-up in the restricted track length. The cart displacement in the restricted track length is verified by simulation and experimental test-run. The regulation problem of stabilization of pendulum is addressed by developing the controller using Pole Placement Controller (PPC) and LQR Controller (LQRC). Both the control strategies are performed analytically and experimentally using the Googoltech Linear Inverted Pendulum (GLIP) setup. The analytical results, simulated in MATLAB and SIMULINK environment, are found in close agreement with the experimental results. In order to demonstrate the effect of both the stabilizing controllers on the performance of the system, comparison of the experimental results is reported in this work. It is demonstrated experimentally that LQR controller outperforms the Pole Placement controller, in terms of reduction in the oscillations of the inverted pendulum (56 %), as well as the magnitude of maximum control input (66.7 %). Further, robustness of the closed-loop system is investigated by providing external disturbances.

Keywords: cart-pendulum system; energy controller; LQR controller; pole placement controller; stabilization; actuator dynamics; robustness

MSC 2010: 70E60; 70Q05; 49N05

Appendix A

The gains for stabilization controller are calculated from eq. (10), which is single pendulum equation of motion. This equation is considered because of stabilization occurs at inverted pendulum equilibrium position. The state-space representation for eq. (10) is given below

z˙t=A1zt+B1ut,

yt=Czt+Dut.

where,

A1=[010000000001003g4l0],[0134l 0]

From Table 1, A1 and B1 can be written as

A1=0100000000010029.40, B1=0103

The eigenvalues of A1 are [5.422, ‒5.422, 0, 0], which shows the system is unstable.

A.1 Pole placement controller

The desired closed-loop poles for Pole Placement controller are selected as [‒10, ‒10, ‒2+j√3, ‒2-j√3]. These poles are selected to achieve the settling time within 2 seconds. Gains for the closed-loop system are calculated as KPPC=[‒54.4218, ‒24.4898, 93.2739, 16.1633]. The choice of the desired poles is justified, as it can be observed in Figure 27 that both the cart as well as pole reach the desired state within 2 seconds. Both analytical and experimental results using Pole Placement controller are obtained for the validation.

Figure 27:

Step response using Pole Placement controller.

A.2 LQR controller

In this section, the control gain matrix is calculated using optimal control theory. Selection of Q and R matrices for LQR controller is done by manual tuning, with the objective of achieving the settling time within 2 seconds. For this Q and R taken as

Q=100000000000020000000, R=1.

Using eq. (36)–eq. (38), the optimal control gain matrix for LQR controller is calculated as KLQRC= [‒31.623, ‒20.151, 72.718, 13.155]. The selection of Q and R is justified by taking step response of the plant with resulted gains. The cart and pole reaches the desired state within 2 seconds, as shown in Figure 28.

Figure 28:

Step response using LQR controller.

References

[1] Y. Liu and H. Yu, A survey of underactuated mechanical systems, IET Control Theory Appl. 7 (2013), 921–935.10.1049/iet-cta.2012.0505Search in Google Scholar

[2] C.A. Ibanez, J.M. Mendoza and J. Davila, Stabilization of the cart pole system: By sliding mode control, Nonlinear Dyn. 78 (2014), 2769–2777.10.1007/s11071-014-1624-6Search in Google Scholar

[3] M. Gulan, M. Salaj and B.R. Ilkiv, Achieving an equilibrium position of pendubot via swing-up and stabilizing model predictive control, J. Electrical Eng. 65 (2014), 356–363.10.2478/jee-2014-0058Search in Google Scholar

[4] X. Lai, J. She, W. Cao and S. Yang, Stabilization of underactuated acrobat based on motion-state constraints, Int. J. Non Linear Mech. 77 (2015), 342–347.10.1016/j.ijnonlinmec.2015.09.006Search in Google Scholar

[5] M.R. Neria, S.H. Ramirez, R.G. Moctezuma and A.L. Juarez, Linear active disturbance rejection control of underactuated systems: The case of the Furuta pendulum, ISA Trans. 53 (2014), 920–928.10.1016/j.isatra.2013.09.023Search in Google Scholar PubMed

[6] R. Kelly and J. Sandoval, Analysis of an estimate of the region of attraction of an IDA-PBC control of a ball and beam underactuated mechanism, Tran. Control Mech. Syst. 2 (2013), 43–48.Search in Google Scholar

[7] S. Zhao, X. Wang, T. Hu, D. Zhang and L. Shen, Flocking control in networks of multiple VTOL agents with nonlinear and under-actuated features, Int. J. Rob. Autom. 6 (2015), 1–6.Search in Google Scholar

[8] P. Mason, M. Broucke and B. Piccoli, Time optimal swing-up of the planar pendulum, IEEE Trans. Automat Contr. 53 (2008), 1876–1886.10.1109/CDC.2007.4434688Search in Google Scholar

[9] J.H. Yang, S. Shim, J.H. Seo and Y.S. Lee, Swing-up control for an inverted pendulum with restricted cart rail length, Int. J. Control Autom. Syst. 7 (2009), 674–680.10.1007/s12555-009-0419-xSearch in Google Scholar

[10] N. Muskinja and B. Tovornik, Swinging up and stabilization of real inverted pendulum, IEEE Trans. Ind. Electron. 53 (2006), 631–639.10.1109/TIE.2006.870667Search in Google Scholar

[11] M.I. Solihin and R. Akmeliawati, Particle swam optimization for stabilizing controller of a self-erecting linear inverted pendulum, Int. J. Electrical Electron. Syst. Res. 3 (2010), 410–145.Search in Google Scholar

[12] N.A. Chaturvedi, N.H. McClamroch and D.S. Bernstein, Stabilization of a 3D axially symmetric pendulum, Automatica 44 (2008), 2258–2265.10.1016/j.automatica.2008.01.013Search in Google Scholar

[13] L.J. Pinto, D.H. Kim, J.Y. Lee and C.S. Han, “Development of a segway robot for an intelligent transport system”, IEEE/SICE International Symposium on System Integration (SII) Kyushu University, Fukuoka, Japan, pp. 710–715, 2012.10.1109/SII.2012.6427308Search in Google Scholar

[14] Y. Daud, A.A. Mamun and J.X. Xu, Dynamic modeling and characteristics analysis of lateral-pendulum unicycle robot, Robotica 35 (2017), 537–568.10.1017/S0263574715000703Search in Google Scholar

[15] P. Olejnik, J. Awrejcewicz and M. Nielaczny, Solution of the Kalman filtering problem in control and modeling of a double inverted pendulum with rolling friction, Pomiary Automatyka Robotyka 17(1) (2013), 63–70.Search in Google Scholar

[16] B. Su, T. Wang, S. Kuang and J. Wang, Effects of physical parameters on dynamic behaviour of ballbot-like robots, Int. J. Rob. Autom. 29 (2014), 105–115.10.2316/Journal.206.2014.1.206-3957Search in Google Scholar

[17] D.E. Orin, A. Goswami and S.H. Lee, Centroidal dynamics of a humanoid robot, Auton. Robot 35 (2013), 161–176.10.1007/s10514-013-9341-4Search in Google Scholar

[18] K.J. Astrom and K. Furuta, Swinging up a pendulum by energy control, Automatica 36 (2000), 287–295.10.1016/S0005-1098(99)00140-5Search in Google Scholar

[19] R. Lozano, I. Fantoni and D.J. Block, Stabilization of the inverted pendulum around its homoclinic orbit, Syst. Control Lett. 40 (2000), 197–204.10.1016/S0167-6911(00)00025-6Search in Google Scholar

[20] D. Chatterjee, A. Patra and H.K. Joglekar, Swing-up and stabilization of a cart-pendulum system under restricted cart track length, Syst. Control Lett. 47 (2002), 355–364.10.1016/S0167-6911(02)00229-3Search in Google Scholar

[21] J.H. Yang, S.Y. Shim, J.H. Seo and Y.S. Lee, Swing-up control for an inverted pendulum with restricted cart rail length, Int. J. Control Autom. Syst. 7 (2009), 674–680.10.1007/s12555-009-0419-xSearch in Google Scholar

[22] B.A. Elsayed, M.A. Hassan and S. Mekhilef, Fuzzy swinging-up with sliding mode control for third order cart-inverted pendulum system, Int. J. Control Autom. Syst. 13 (2015), 238–248.10.1007/s12555-014-0033-4Search in Google Scholar

[23] W. Li, H. Ding and K. Cheng, “An investigation on the design and performance assessment of double-PID and LQR controllers for the inverted pendulum”, UKACC International Conference on Control, 2012.10.1109/CONTROL.2012.6334628Search in Google Scholar

[24] W.D. Chang, R.C. Hwang and J.G. Hsieh, A self-tuning PID control for a class of nonlinear systems based on the Lyapunov approach, J. Process Control. 12 (2002), 233–242.10.1016/S0959-1524(01)00041-5Search in Google Scholar

[25] V. Kumar and J. Jerome, Robust LQR controller design for stabilizing and trajectory tracking of inverted pendulum, Procedia Eng. 64 (2013), 169 – 178.10.1016/j.proeng.2013.09.088Search in Google Scholar

[26] K. Ogata, Modern control engineering, Prentice-Hall, New York, 2000.Search in Google Scholar

[27] S. Mori, H. Nishihara and K. Furuta, Control of unstable mechanical system‐ control of pendulum, Int. J. Control 23 (1976), 673 – 692.10.1080/00207177608922192Search in Google Scholar

[28] C.W. Anderson, Learning to control an inverted pendulum using neural network, IEEE Control Syst. Mag. 9 (1989), 31–37.10.1109/37.24809Search in Google Scholar

[29] O. Boubaker, The inverted pendulum benchmark in nonlinear control theory, Int. J. Adv. Rob. Syst. 10 (2013), 1–9.10.5772/55058Search in Google Scholar

[30] Z. Song, X. Song, C. Liu and Y. Zhao, Research on real-time simulation and control of linear 1-stage inverted pendulum, J. Computers 8 (2013), 896–902.10.4304/jcp.8.4.896-903Search in Google Scholar

[31] R. Eide, M. Egelid, A. Stamso and H.R. Karimi, LQG control design for balancing an inverted pendulum mobile robot, Intell. Control Autom. 2 (2011), 160–166.10.4236/ica.2011.22019Search in Google Scholar

[32] B.C. Kuo and G. Golnaraghi, Automatic control systems, 8^th edition, Wiley, Hoboken, NJ, 2003.Search in Google Scholar

[33] A. Tewari, Modern control design with MATLAB & SIMULINK, Wiley, Chichester, 2002.Search in Google Scholar

[34] M.E. Magana, Fuzzy-logic control of an inverted pendulum with vision feedback, IEEE Trans. Educ. 41 (1998), 165–170.10.1109/13.669727Search in Google Scholar

[35] H. Schmirgel, J.O. Krah and R. Berger, Delay time compensation in the current control loop of servo drives – higher bandwidth at no trade-off, Proceedings of PCIM Power conversion and Intelligent Motion, 541–546, 2006.Search in Google Scholar

Received: 2017-2-11

Revised: 2017-7-11

Accepted: 2017-8-7

Published Online: 2017-9-14

Published in Print: 2017-10-26

You are currently not able to access this content.

Articles in the same Issue

https://doi.org/10.1515/ijnsns-2017-0040

Keywords for this article

cart-pendulum system; energy controller; LQR controller; pole placement controller; stabilization; actuator dynamics; robustness