##
**Delay-dependent non-synchronized robust \({\mathcal H}_\infty\) state estimation for discrete-time piecewise linear delay systems.**
*(English)*
Zbl 1191.93132

Summary: This paper investigates the problem of delay-dependent robust \({\mathcal H}_\infty\) filtering design for a class of uncertain discrete-time piecewise linear state-delayed systems where state space instead of measurable output space partitions are assumed so that filter implementation may not be synchronized with plant state trajectory transitions. The state delay is assumed to be time-varying and of an interval-like type. The uncertainties are assumed to have a structured linear fractional form. The objective is to design a piecewise linear state estimator guaranteeing the asymptotic stability of the resulting filtering error system with robust \({\mathcal H}_\infty\) performance \(\gamma \). Based on a new delay-dependent piecewise Lyapunov-Krasovskii functional combined with Finsler’s Lemma, a novel delay-dependent robust \({\mathcal H}_\infty\) performance analysis result is first presented and the filter synthesis is then developed. It is shown that the filter gains can be obtained by solving a set of linear matrix inequalities, which are numerically efficient with commercially available software. Finally, a numerical example is provided to illustrate the effectiveness and less conservatism of the proposed approach.

### MSC:

93E10 | Estimation and detection in stochastic control theory |

93C55 | Discrete-time control/observation systems |

93B36 | \(H^\infty\)-control |

### Keywords:

piecewise linear systems; robust \(\mathcal H_{\infty }\) filtering; linear fractional uncertainty; time-varying delay; delay-dependent; linear matrix inequalities### Software:

LMI toolbox
PDF
BibTeX
XML
Cite

\textit{J. Qiu} et al., Int. J. Adapt. Control Signal Process. 23, No. 12, 1082--1096 (2009; Zbl 1191.93132)

Full Text:
DOI

### References:

[1] | Kalman, A new approach to linear filtering and prediction problems, Transactions of the ASMEâJournal of Basic Engineering 82D (1) pp 35– (1960) |

[2] | Anderson, Optimal Filtering (1979) |

[3] | Zhu, Design and analysis of discrete-time robust Kalman filters, Automatica 38 (6) pp 1069– (2002) · Zbl 1001.93078 |

[4] | Garcia, Robust Kalman filtering for uncertain discrete-time linear systems, International Journal of Robust and Nonlinear Control 13 (13) pp 1225– (2003) · Zbl 1039.93062 |

[5] | Li, A linear matrix inequalities approach to robust ââ filtering, IEEE Transactions on Signal Processing 45 (9) pp 2338– (1998) |

[6] | Geromel, â2 and ââ robust filtering for convex bounded uncertain systems, IEEE Transactions on Automatic Control 46 (1) pp 100– (2000) |

[7] | Tuan, Robust and reduced-order filtering: new LMI-based characterizations and methods, IEEE Transactions on Signal Processing 49 (12) pp 2975– (2001) |

[8] | Gao, New approach to mixed â2/ââ filtering for polytopic discrete-time systems, IEEE Transactions on Signal Processing 53 (8) pp 3183– (2005) |

[9] | Zhang, Robust ââ filtering for switched linear discrete-time systems with polytopic uncertainties, International Journal of Adaptive Control and Signal Processing 20 (6) pp 291– (2006) |

[10] | Johansson, Computation of piecewise Lyapunov quadratic functions for hybrid systems, IEEE Transactions on Automatic Control 43 (4) pp 555– (1998) · Zbl 0905.93039 |

[11] | Rantzer, Piecewise linear quadratic optimal control, IEEE Transactions on Automatic Control 45 (4) pp 629– (2000) · Zbl 0969.49016 |

[12] | Cuzzola, An LMI approach for ââ analysis and control of discrete-time piecewise affine systems, International Journal of Control 75 (16â17) pp 1293– (2002) · Zbl 1015.93015 |

[13] | Feng, Stability analysis of piecewise discrete-time linear systems, IEEE Transactions on Automatic Control 47 (7) pp 1108– (2002) · Zbl 1364.93563 |

[14] | Rodrigues L, How JP. Observer-based control of piecewise-affine systems. Proceedings of the 40th IEEE Conference on Decision and Control, Orlando, FL, U.S.A., 2001; 1366â1371. |

[15] | Saito, Control of chaos from a piecewise-linear hysteresis circuit, IEEE Transactions on Circuits and SystemsâPart I: Fundamental Theory and Applications 41 (3) pp 168– (1995) |

[16] | Storace, Piecewise-linear approximation of nonlinear dynamical systems, IEEE Transactions on Circuits and SystemsâPart I: Regular Papers 51 (4) pp 830– (2004) · Zbl 1374.37069 |

[17] | Ferrari-Trecate, Moving horizon estimation for hybrid systems, IEEE Transactions on Automatic Control 47 (2) pp 1663– (2002) · Zbl 1364.93768 |

[18] | Feng, Robust filtering design of piecewise discrete time linear systems, IEEE Transactions on Signal Processing 53 (2) pp 599– (2005) · Zbl 1370.93273 |

[19] | Feng, Non-synchronized state estimation of discrete time piecewise linear systems, IEEE Transactions on Signal Processing 54 (1) pp 295– (2006) |

[20] | Xu J, Xie L. Non-synchronized ââ estimation of piecewise linear systems. Proceedings of the 1st IEEE Conference on Industrial Electronics and Applications, Singapore, 2006; 1â6. |

[21] | Juloski AL, Heemels WPMH, Boers Y, Verschure F. Two approaches to state estimation for a class of piecewise affine systems. Proceedings of the 42nd IEEE Conference on Decision and Control, Hawaii, U.S.A., 2003; 143â148. |

[22] | Xu J, Xie L. ââ estimation of discrete-time piecewise linear systems. Proceedings of the 8th International Conference on Control, Automation, Robotics and Vision, Kunming, China, 2004; 1163â1168. |

[23] | Palhares, Robust ââ filtering for uncertain discrete-time state-delayed systems, IEEE Transactions on Signal Processing 49 (8) pp 1696– (2001) |

[24] | Gao, A delay-dependent approach to robust ââ filtering for uncertain discrete-time state delayed systems, IEEE Transactions on Signal Processing 52 (6) pp 1631– (2004) · Zbl 1052.93056 |

[25] | Fridman, Robust ââ filtering of linear systems with time-varying delay, IEEE Transactions on Automatic Control 48 (1) pp 159– (2003) |

[26] | Xu, Robust ââ control for uncertain discrete-time systems with time-varying delays via exponential output feedback controllers, Systems and Control Letters 51 (3â4) pp 171– (2004) · Zbl 1157.93371 |

[27] | Wang, Robust ââ filtering for stochastic time-delay systems with missing measurements, IEEE Transactions on Signal Processing 54 (7) pp 2579– (2006) |

[28] | Hu, Delay-dependent filtering design for time-delay systems with Markovian jumping parameters, International Journal of Adaptive Control and Signal Processing 21 (5) pp 434– (2007) · Zbl 1120.60043 |

[29] | Zhang, Delay-dependent robust ââ filtering for uncertain discrete-time systems with time-varying delay based on a finite sum inequality, IEEE Transactions on Circuits and SystemsâPart II: Express Briefs 53 (12) pp 1466– (2006) |

[30] | Gao, Discrete bilinear stochastic systems with time-varying delay: stability analysis and control synthesis, Chaos, Solitons and Fractals 2 (34) pp 394– (2007) · Zbl 1134.93413 |

[31] | Jiang, Robust Hâ control for uncertain Takagi-Sugeno fuzzy systems with interval time-varying delay, IEEE Transactions on Fuzzy systems 15 (2) pp 321– (2007) |

[32] | Chakrabarty, Control of chaos in piecewise linear systems with switching nonlinearity, Physics Letters A 200 (2) pp 115– (1995) |

[33] | Zhang, Robust ââ filtering for uncertain discrete-time piecewise time-delay systems, International Journal of Control 80 (4) pp 636– (2007) |

[34] | Qiu, New results on robust ââ filtering design for discrete piecewise linear delay systems, International Journal of Control 82 (1) pp 183– (2009) |

[35] | Boyd, Linear Matrix Inequality in Systems and Control Theory (1994) · Zbl 0816.93004 |

[36] | Gahinet, LMI Control Toolbox User’s Guide (1995) |

[37] | Gao, New results on stability of discrete-time systems with time-varying state delays, IEEE Transactions on Automatic Control 52 (2) pp 328– (2007) |

[38] | He, Delay-range-dependent stability for systems with time-varying delay, Automatica 43 (2) pp 371– (2007) · Zbl 1111.93073 |

[39] | He Y, Wu M, Liu GP, She JH. Output feedback stabilization for discrete-time systems with a time-varying delay. Proceedings of the 26th Chinese Control Conference, Zhangjiajie, Hunan, China, 2007; 64â70. |

[40] | Yue, State feedback controller design of networked control systems, IEEE Transactions on Circuits and SystemsâPart II: Express Briefs 51 (11) pp 640– (2004) |

[41] | Yang, ââ control for networked systems with random communication delays, IEEE Transactions on Automatic Control 51 (3) pp 511– (2006) |

[42] | Gao, A new delay system approach to network-based control, Automatica 44 (1) pp 39– (2008) · Zbl 1138.93375 |

[43] | Gao, ââ estimation for uncertain systems with limited communication capacity, IEEE Transactions on Automatic Control 52 (11) pp 2070– (2007) |

[44] | Xie, Output feedback control of systems with parameter uncertainty, International Journal of Control 63 (4) pp 741– (1996) · Zbl 0841.93014 |

[45] | Zhou, New characterization of positive realness and control of a class of uncertain polytopic discrete-time systems, Systems and Control Letters 54 (5) pp 417– (2005) · Zbl 1129.93441 |

[46] | Ghaoui, Control of rational systems using linear-fractional representations and linear matrix inequalities, Automatica 32 (9) pp 1273– (1996) · Zbl 0857.93040 |

[47] | Skelton, A Unified Algebraic Approach to Linear Control Design (1998) |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.