Equal Ratio Gain Technique and Its Application in Linear General Integral Control

In conjunction with linear general integral control, this paper proposes a fire-new control design technique, named Equal ratio gain technique, and then develops two kinds of control design methods, that is, Decomposition and Synthetic methods, for a class of uncertain nonlinear system. By Routh’s stability criterion, we demonstrate that a canonical system matrix can be designed to be always Hurwitz as any row controller gains, or controller and its integrator gains increase with the same ratio. By solving Lyapunov equation, we demonstrate that as any row controller gains, or controller and its integrator gains of a canonical system matrix tend to infinity with the same ratio, if it is always Hurwitz, and then the same row solutions of Lyapunov equation all tend to zero. By Equal ratio gain technique and Lyapunov method, theorems to ensure semi-globally asymptotic stability are established in terms of some bounded information. Moreover, the striking robustness of linear general integral control and PID control is clearly illustrated by Equal ratio gain technique. Theoretical analysis, design example and simulation results showed that Equal ratio gain technique is a powerful tool to solve the control design problem of uncertain nonlinear system.

Share and Cite:

Liu, B. (2015) Equal Ratio Gain Technique and Its Application in Linear General Integral Control. International Journal of Modern Nonlinear Theory and Application, 4, 21-36. doi: 10.4236/ijmnta.2015.41003.

Conflicts of Interest

The authors declare no conflicts of interest.

 [1] Khalil, H.K. (2007) Nonlinear Systems. 3rd Edition, Electronics Industry Publishing, Beijing, 449-453, 551. [2] Liu, B.S. and Tian, B.L. (2012) General Integral Control Design Based on Linear System Theory. Proceedings of the 3rd International Conference on Mechanic Automation and Control Engineering, 5, 3174-3177. [3] Liu, B.S. and Tian, B.L. (2012) General Integral Control Design Based on Sliding Mode Technique. Proceedings of the 3rd International Conference on Mechanic Automation and Control Engineering, 5, 3178-3181. [4] Liu, B.S., Li, J.H. and Luo, X.Q. (2014) General Integral Control Design via Feedback Linearization. Intelligent Control and Automation, 5, 19-23. http://dx.doi.org/10.4236/ica.2014.51003 [5] Liu, B.S., Luo, X.Q. and Li, J.H. (2014) General Integral Control Design via Singular Perturbation Technique. International Journal of Modern Nonlinear Theory and Application, 3, 173-181. http://dx.doi.org/10.4236/ijmnta.2014.34019 [6] Liu, B.S., Luo, X.Q. and Li, J.H. (2013) General Concave Integral Control. Intelligent Control and Automation, 4, 356-361. http://dx.doi.org/10.4236/ica.2013.44042 [7] Liu, B.S., Luo, X.Q. and Li, J.H. (2014) General Convex Integral Control. International Journal of Automation and Computing, 11, 565-570. http://dx.doi.org/10.1007/s11633-014-0813-6 [8] Liu, B.S. (2014) Constructive General Bounded Integral Control. Intelligent Control and Automation, 5, 146-155. http://dx.doi.org/10.4236/ica.2014.53017 [9] Liu, B.S. (2014) On the Generalization of Integrator and Integral Control Action. International Journal of Modern Nonlinear Theory and Application, 3, 44-52. http://dx.doi.org/10.4236/ijmnta.2014.32007 [10] Gajic, Z. (1995) Lyapunov Matrix Equation in System Stability and Control. Mathematics in Science and Engineering, 195, 30-31.