Finite-time functional interval observer for linear systems with uncertainties. (English) Zbl 07916986
Summary: This study investigates the problem of designing a finite-time functional interval observer (FTFIO) for linear systems with uncertainties. A novel structure is proposed to design an FTFIO for uncertain systems. Based on the proposed structure, the sufficient conditions for the existence of such a finite-time observer are established as a set of coefficient matrix equations with a constraint matrix. Meanwhile, using the obtained existence conditions, the parametric expression of an FTFIO is developed by adopting the solutions to a type of first-order Sylvester equations, which provides extra degrees of freedom. Finally, the FTFIO design method is efficiently applied to the estimation problem in two examples, including the simple numerical system and chaotic system.
© 2021 The Authors. IET Control Theory & Applications published by John Wiley & Sons, Ltd. on behalf of The Institution of Engineering and Technology
© 2021 The Authors. IET Control Theory & Applications published by John Wiley & Sons, Ltd. on behalf of The Institution of Engineering and Technology
MSC:
| 93B53 | Observers |
| 93C05 | Linear systems in control theory |
| 93C41 | Control/observation systems with incomplete information |
Keywords:
control system synthesis; observers; linear systems; uncertain systems; matrix algebra; coefficient matrix equations; FTFIO design method; chaotic system; finite-time functional interval observer; linear systems; uncertain systems; finite-time observer; first-order Sylvester equationsReferences:
| [1] | DimassiH.WinkinJ.J.WouwerA.V.: ‘A sliding mode observer for a linear reaction-convection-diffusion equation with disturbances’, Syst. Control Lett., 2019, 124, pp. 40-48 · Zbl 1408.93038 |
| [2] | BezzaouchaS.VoosH.DarouachM.: ‘A new polytopic approach for the unknown input functional observer design’, Int. J. Control, 2018, 91, (3), pp. 658-677 · Zbl 1396.93028 |
| [3] | OrtegaR.SarrA.BobtsovA. et al.: ‘Adaptive state observers for sensorless control of switched reluctance motors’, Int. J. Robust Nonlinear Control, 2019, 29, (4), pp. 990-1006 · Zbl 1418.93122 |
| [4] | KlineM.: ‘Mathematics: the loss of certainty’ (Oxford University Press, London, UK, 1982) |
| [5] | LiuL.W.GuD.K.: ‘Robust \(H_{\operatorname{\infty}}\) interval observer for linear systems with a controllable convergence rate: a parametric method’, IEEE Access, 2019, 7, pp. 123922-123933 |
| [6] | GouzéJ.L.RapaportA.Hadj‐SadokM.Z.: ‘Interval observers for uncertain biological systems’, Ecol. model., 2000, 133, (1‐2), pp. 45-56 |
| [7] | MoisanM.BernardO.: ‘Interval observers for non‐monotone systems. Application to bioprocess models’, IFAC Proc. Vol., 2005, 38, (1), pp. 43-48 |
| [8] | RaïssiT.EfimovD.ZolghadriA.: ‘Interval state estimation for a class of nonlinear systems’, IEEE Trans. Autom. Control, 2011, 57, (1), pp. 260-265 · Zbl 1369.93074 |
| [9] | EfimovD.PerruquettiW.RaissiT. et al.: ‘On IO design for time‐invariant discrete‐time systems’. Proc. IEEE on European Control Conf., Zurich, Switzerland, July 2013, pp. 2651-2656 |
| [10] | CacaceF.GermaniA.ManesC.: ‘A new approach to design interval observers for linear systems’, IEEE Trans. Autom. Control, 2014, 60, (6), pp. 1665-1670 · Zbl 1360.93117 |
| [11] | WangY.BevlyD.M.RajamaniR.: ‘Interval observer design for LPV systems with parametric uncertainty’, Automatica, 2015, 60, pp. 79-85 · Zbl 1331.93033 |
| [12] | FaraminM.RezaieB.RahmaniZ.: ‘Robust integral feedback control based on interval observer for stabilising parameter‐varying systems’, IET Control Theory Appl., 2019, 13, (15), pp. 2455-2464 |
| [13] | ThabetR.E.H.RaissiT.CombastelC. et al.: ‘An effective method to interval observer design for time‐varying systems’, Automatica, 2014, 50, (10), pp. 2677-2684 · Zbl 1301.93035 |
| [14] | ZhuJ.JohnsonC.D.: ‘Unified canonical forms for matrices over a differential ring’, Linear Algebra Appl., 1991, 147, pp. 201-248 · Zbl 0719.15004 |
| [15] | PourasgharM.PuigV.Ocampo‐MartinezC. et al.: ‘Reduced‐order interval‐observer design for dynamic systems with time‐invariant uncertainty’, IFAC‐PapersOnLine, 2017, 50, (1), pp. 6271-6276 |
| [16] | GuD.K.LiuL.W.DuanG.R.: ‘Functional interval observer for the linear systems with disturbances’, IET Control Theory Appl., 2018, 12, (18), pp. 2562-2568 |
| [17] | GuD.K.LiuL.W.DuanG.R.: ‘Design of reduced‐order interval observers for LTI systems with bounded time‐varying disturbances: a parametric approach’, Int. J. Control, 2019, pp. 1-27, published online, doi: https://doi.org/10.1080/00207179.2019.1675901 · Zbl 1454.93029 · doi:10.1080/00207179.2019.1675901 |
| [18] | HuongD.C.: ‘Interval observers for linear functions of state vectors of linear fractional‐order systems with delayed input and delayed output’, Int. J. Adapt. Control Signal Process., 2019, 33, (3), pp. 445-461 · Zbl 1417.93149 |
| [19] | EthabetH.RabehiD.EfimovD. et al.: ‘Interval estimation for continuous‐time switched linear systems’, Automatica, 2018, 90, pp. 230-238 · Zbl 1387.93152 |
| [20] | ZhengG.EfimovD.BejaranoF.J. et al.: ‘Interval observer for a class of uncertain nonlinear singular systems’, Automatica, 2016, 71, pp. 159-168 · Zbl 1343.93022 |
| [21] | PangB.ZhangQ.: ‘Interval observers design for polynomial fuzzy singular systems by utilizing sum‐of‐squares program’, IEEE Trans. Syst. Man Cybern., Syst., 2020, 50, (6), pp. 1999-2006, published online, doi: https://doi.org/10.1109/TSMC.2018.2790975 · doi:10.1109/TSMC.2018.2790975 |
| [22] | ThabetR.E.H.Ahmed AliS.PuigV.: ‘High‐gain interval observer for partially linear systems with bounded disturbances’, Int. J. Control, 2019, pp. 1-10, published online, doi: https://doi.org/10.1080/00207179.2019.1650204 · Zbl 1472.93058 · doi:10.1080/00207179.2019.1650204 |
| [23] | LiJ.WangZ.ShenY. et al.: ‘Interval observer design for discrete‐time uncertain Takagi+Sugeno fuzzy systems’, IEEE Trans. Fuzzy Syst., 2019, 27, (4), pp. 816-823 |
| [24] | ElleroN.Gucik‐DerignyD.HenryD.: ‘Unknown input interval observer with \(H_{\operatorname{\infty}}\) and \(D\) ‐stability performance’, IFAC‐PapersOnLine, 2017, 50, (1), pp. 6251-6258 |
| [25] | MazencF.AhmedS.MalisoffM.: ‘Finite time estimation through a continuous‐discrete observer’, Int. J. Robust Nonlinear Control, 2018, 28, (16), pp. 4831-4849 · Zbl 1402.93236 |
| [26] | EngelR.KreisselmeierG.: ‘A continuous‐time observer which converges in finite time’, IEEE Trans. Autom. Control, 2002, 47, (7), pp. 1202-1204 · Zbl 1364.93084 |
| [27] | ZhangJ.ChadliM.ZhuF.: ‘Finite‐time observer design for singular systems subject to unknown inputs’, IET Control Theory Appl., 2019, 13, (14), pp. 2289-2299 · Zbl 07907025 |
| [28] | DinhT.N.MazencF.WangZ.H. et al.: ‘On Fixed‐time interval estimation of discrete‐time nonlinear time‐varying systems with disturbances’. Proc. IEEE on American Control Conf., Denver, CO, USA, July 2020, hal‐02493776 |
| [29] | PerruquettiW.FloquetT.MoulayE.: ‘Finite‐time observers: application to secure communication’, IEEE Trans. Autom. Control, 2008, 53, (1), pp. 356-360 · Zbl 1367.94361 |
| [30] | DuH.QianC.YangS. et al.: ‘Recursive design of finite‐time convergent observers for a class of time‐varying nonlinear systems’, Automatica, 2013, 49, (2), pp. 601-609 · Zbl 1259.93029 |
| [31] | EfimovT.RaïssiD.ChebotarevS. et al.: ‘Interval state observer for nonlinear time varying systems’, Automatica, 2013, 49, (1), pp. 200-205 · Zbl 1258.93032 |
| [32] | KaczorekT.: ‘Positive 1D and 2D systems’ (Springer Science & Business Media, London, 2012) |
| [33] | DuanG.R.: ‘Generalized Sylvester equations: unified parametric solutions’ (CRC Press, Boca Raton, FL, 2015) · Zbl 1329.15006 |
| [34] | ZhouB.DuanG.R.: ‘A new solution to the generalized Sylvester matrix equation AV‐EVF = BW’, Syst. Control Lett., 2006, 55, (3), pp. 193-198 · Zbl 1129.15300 |
| [35] | MaX.HuangJ.ChenL.: ‘Finite‐time interval observers‐design for switched systems’, Circuits, Syst. Signal Process., 2019, 38, (11), pp. 5304-5322 |
| [36] | MeyerL.: ‘Robust functional interval observer for multivariable linear systems’, J. Dyn. Syst. Meas. Control, 2019, 141, (9), pp. 094502‐1-094502‐7 |
| [37] | DieciL.EirolaT.: ‘On smooth decomposition of matrices’, SIAM J. Matrix Anal. Appl., 1999, 20, (3), pp. 800-819 · Zbl 0930.15014 |
| [38] | MoisanM.BernardO.: ‘Robust interval observers for global Lipschitz uncertain chaotic systems’, Syst. Control Lett., 2010, 59, (11), pp. 687-694 · Zbl 1216.34050 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
