The paper is concerned with finite-time L1 control for positive Markovian jump systems with constant time delay and partly known transition rates. By constructing appropriate co-positive type Lyapunov function, sufficient conditions for finite-time boundedness and L1 finite-time boundedness of the underlying system are firstly given. Based on the results obtained, a state feedback controller is designed such that the closed-loop Markovian jump system is positive and L1 finite-time bounded. All the proposed conditions are given in linear programming. Finally, an example is given to demonstrate the validity of the main results.
Ait RamiMTadeoF (2007) Controller synthesis for positive linear systems with bounded controls. IEEE Transactions on Circuits and Systems Part II: Express Briefs54(2): 151–155.
2.
BolzernPColaneriPNicolaoG (2014) Stochastic stability of positive Markov jump linear systems. Automatica50(4): 1181–1187.
3.
CaccettaLFouldsLRRumchevVG (2004) A positive linear discrete-time model of capacity planning and its controllability properties. Mathematical and Computer Modelling40(1–2): 217–226.
4.
ChenX MJamesLLiP. (2013) l1-induced norm and controller synthesis of positive systems. Automatica49(5): 1377–1385.
5.
ChenGPYangY (2014) Finite-time stability of switched positive linear systems. International Journal of Robust and Nonlinear Control24(1): 179–190.
6.
EbiharaYPeaucelleDArzelierD (2014) LMI approach to linear positive system analysis and synthesis. Systems and Control Letters63: 50–56.
7.
FarinaLRinaldiS (2000) Positive Linear Systems: Theory and Applications. New York: Wiley.
8.
FornasiniEValcherME (2012) Stability and stabilizability criteria for discrete-time positive switched systems. IEEE Transactions on Automatic Control57(5): 1208–1221.
9.
GeXHHanQL (2014) Distributed fault detection over sensor networks with Markovian switching topologies. International Journal of General Systems43(3–4): 305–318.
10.
KaczorekT (2002) Positive 1D and 2D Systems. Springer-Verlag.
11.
KaoYGXieJWangCH (2014) Stabilisation of singular Markovian jump systems with generally uncertain transition rates. IEEE Transactions on Automatic Control59(9): 2604–2610.
12.
KaoYGXieJWangCH. (2015) A sliding mode approach to H∞ non-fragile observer-based control design for uncertain Markovian neutral-type stochastic systems. Automatica52: 218–226.
13.
LiYBKaoYGXieJ (2014) Robust guaranteed cost observer design for singular Markovian jump time-delay systems with generally incomplete transition probability. Abstract and Applied Analysis. 2014: 1–11.
14.
LiPLamJShuZ (2010) H∞ positive filtering for positive linear discrete-Time systems: an augmentation approach. IEEE Transactions on Automatic Control55(10): 2337–2342.
15.
LiSXiangZRKarimiHR (2013) Stability and L1-gain controller design for positive switched systems with mixed time-varying delays. Applied Mathematics and Computation222: 507–518.
16.
LiuXDangC (2011) Stability analysis of positive switched linear systems with delays. IEEE Transactions on Automatic Control56(7): 1684–1690.
17.
ShenLJBuscherU (2012) Solving the serial batching problem in job shop manufacturing systems. European Journal of Operational Research221(1): 14–26.
18.
ShortenRWirthFLeithD (2006) A positive systems model of TCP-like congestion control: asymptotic results. IEEE/ACM Transactions on Networking14(3): 616–629.
19.
SongYXieJXFeiMR. (2013) Mean square exponential stabilization of networked control systems with Markovian packet dropouts. Transactions of the Institute of Measurement and Control35(1): 75–82.
20.
SunHHovakimyanNBasarT (2012) L1 Adaptive controller for uncertain nonlinear multi-input multi-output systems with input quantization. IEEE Transactions on Automatic Control57(3): 565–578.
21.
XiangMXiangZR (2013) Finite-time L1 control for positive switched linear systems with time-varying delay. Communications in Nonlinear Science and Numerical Simulation18(11): 3158–3166.
22.
XuSYChenTW (2003) Robust H-infinity filtering for uncertain stochastic time-delay systems. Asian Journal of Control5(3): 364–373.
23.
XuSYShiPChuYM. (2006) Robust stochastic stabilization and H-infinity control of uncertain neutral stochastic time-delay systems. Journal of Mathematical Analysis and Applications314(1): 1–16.
24.
ZhangJHanZZWuH (2014a) Robust finite-time stability and stabilisation of switched positive systems. IET Control Theory and Applications8(1): 67–75.
25.
ZhangJFHanZZZhuF (2014b) Stochastic stability and stabilization of positive systems with Markovian jump parameters. Nonlinear Analysis: Hybrid Systems12: 147–155.
26.
ZhangJFHanZZZhuF (2014c) Finite-time control and L1-gain analysis for positive switched systems. Optimal Control Applications and Methods36(3).
27.
ZhuSQHanQLZhangCH (2014) l1-gain performance analysis and positive filter design for positive discrete-time Markov jump linear systems: A linear programming approach. Automatica50(8): 2098–2107.
