LMI-based sufficient conditions for robust stability and stabilization of LTI-fractional-order systems subjected to interval and polytopic uncertainties
Restricted accessResearch articleFirst published online November, 2015
LMI-based sufficient conditions for robust stability and stabilization of LTI-fractional-order systems subjected to interval and polytopic uncertainties
In this paper, by introducing a new general state-space form for uncertain linear time-invariant fractional-order systems subjected to interval and polytopic uncertainties, two problems including robust stability analysis and robust stabilization of the presented systems are investigated. Subsequently, two sufficient conditions in terms of several linear matrix inequalities for the problems mentioned are concluded as two separate theorems. It is assumed that the fractional order is a known constant belonging to . Simulation results of two different numerical examples demonstrate that the provided sufficient conditions are applicable and effective for tackling robust stability and stabilization problems.
AguiarBGonzálezTBernalM (2014) Comments on “Robust stability and stabilization of fractional-order interval systems with the fractional order : the case”. IEEE Transactions on Automatic Control DOI: 10.1109/TAC.2014.2332711.
2.
AhnHSChenYQ (2008) Necessary and sufficient stability condition of fractional-order interval linear systems. Automatica44: 2985–2988.
3.
AhnHSChenYQPodlubnyI (2007) Robust stability test of a class of linear time-invariant interval fractional-order system using Lyapunov inequality. Applied Mathematics and Computation187: 27–34.
4.
Akbari-MoornaniKHaeriM (2009) On robust stability of linear time invariant fractional-order systems with real parametric uncertainties. ISA Transactions45: 484–490.
5.
Akbari-MoornaniKHaeriM (2010) Robust stability testing function and Kharitonov-like theorem for fractional order interval systems. IET Control Theory and Application4: 2097–2108.
6.
Akbari-MoornaniKHaeriM (2012) Robust stability check for fractional PID-based control system. Transactions of the Institute of Measurement and Control35: 236–246.
7.
ChenYQAhnHSPodlubnyI (2006) Robust stability check of fractional order linear time invariant systems with interval uncertainties. Signal Processing86: 2611–2618.
8.
EfeMÖ (2010) Integral sliding mode control of a quadrotor with fractional order reaching dynamics. Transactions of the Institute of Measurement and Control33: 985–1003.
9.
FargesCMozeMSabatierJ (2010) Pseudo-state feedback stabilization of commensurate fractional order systems. Automatica46: 1730–1734.
10.
FargesCSabatierJMozeM (2011) Robust stability analysis and stabilization of fractional order polytopic systems. In: Proceedings of the 18th IFAC World Congress, Milano, Italy, September, pp. 10800–10805.
11.
GabanoJDPoinotTKanounH (2011) Identification of a thermal system using continuous linear parameter-varying fractional modeling. IET Control Theory and Applications5: 889–899.
12.
JiaoZZhongY (2012) Robust stability for fractional-order systems with structured and unstructured uncertainties. Computers & Mathematics with Applications64: 3258–3266.
13.
LiCWangJ (2012) Robust stability and stabilization of fractional order interval systems with coupling relationships: The case. Journal of the Franklin Institute349: 2406–2419.
14.
LiaoZPengCLiW. (2011) Robust stability analysis for a class of fractional order systems with uncertain parameters. Journal of the Franklin Institute348: 1101–1113.
15.
LuJGChenG (2009) Robust stability and stabilization of fractional-order interval systems: an LMI approach. IEEE Transactions on Automatic Control54: 1294–1299.
16.
LuJGChenYQ (2010) Robust stability and stabilization of fractional-order interval systems with the fractional order : The case. IEEE Transactions on Automatic Control55: 152–158.
17.
LuJGChenYQ (2013) Stabilization and stabilization of fractional-order linear systems with convex polytopic uncertainties. Fractional Calculus and Applied Analysis16: 142–157.
18.
LuJGChenYQChenW (2013) Robust asymptotical stability of fractional-order linear systems with structured perturbations. Computers & Mathematics with Applications66: 873–882.
19.
MatignonD (1996) Stability result on fractional differential equations with applications to control processing. In: Proceedings of the Computational Engineering in Systems and Application Multi Conference IMACS IEEE-SMC, Lille, France, July, pp. 963–968.
20.
MatignonD (1998) Stability properties for generalized fractional differential systems. Fractional Differential Systems: Models, Methods and Applications5: 145–158.
21.
MatignonDD’Andréa NovelB (1996) Some results on controllability and observability of finite-dimensional fractional differential systems. In: Proceedings of the Computational Engineering in Systems and Application Multi Conference: IMACS IEEE-SMC, Lille, France, July, pp. 952–956.
22.
MatignonDD’Andréa NovelB (1997) Observer-based controllers for fractional differential systems. In: Proceedings of the 36th IEEE Conference on Decision and Control, San Diego, CA, December, pp. 4967–4972.
23.
MillerKSRossB (1993) An Introduction to the Fractional Calculus and Fractional Differential Equations. New York: Wiley.
24.
PodlubnyI (1999) Fractional Differential Equations. Mathematics in Science and Engineering. New York: Academic Press.
25.
SabatierJMozeMFargesC (2010) LMI stability conditions for fractional order systems. Computers & Mathematics with Applications59: 1594–1609.
26.
Tavakoli-KakhkiMHaeriMTavazoeiMS (2010) Over- and under-convergent step responses in fractional-order transfer functions. Transactions of the Institute of Measurement and Control32: 376–394.
