To estimate validated bounds on the actual state vector of uncertain non-linear systems, cooperative output injections methods are proposed in this contribution. The aim of the output injections is to design set-membership state estimators that ensure the order-preserving property between the lower, actual and upper state trajectories. Based on a special sensors placement, continuous-time and event-triggered output injections are proposed to cope with the conservatism of the classical bounding system methods. Furthermore, based on some properties of monotone dynamical systems, the convergence of the proposed set-membership state estimators is shown. It is worth pointing out that the proposed set-membership state estimation method allows one: on one hand, to avoid the conservatism related to the use of similarity transformations usually required in the framework of interval observer design approaches, and on the other hand, to circumvent the pessimism accumulation related to the wrapping effect of set-valued iterative numerical schemes.
KurzhanskiABVályiI.Ellipsoidal calculus for estimation and control. Boston, MA: Birkhaüser, 1996.
2.
BlanchiniF.Set invariance in control. Automatica1999; 35: 1747–1767.
3.
JaulinLKiefferMDidritO, et al. Applied interval analysis: with examples in parameter and state estimation, robust control and robotics. London: Springer-Verlag, 2001.
4.
AsarinEDangTMalerO.D/dt: a tool for reachability analysis of continuous and hybrid systems. IFAC Proc Vol2001; 34(6): 741–746.
5.
BernardOGouzéJ.Closed loop observers bundle for uncertain biotechnological models. J Process Contr2004; 14: 765–774.
6.
ChernouskoF.Ellipsoidal state estimation for dynamical systems. Nonlinear Anal2005; 63: 872–879.
7.
PuigV.Fault diagnosis and fault tolerant control using set-membership approaches: application to real case studies. Int J Appl Math Comp2010; 20: 619–635.
8.
GirardAGuernicCL.Zonotope/hyperplane intersection for hybrid systems reachability analysis. In: EgerstedtMMishraB (eds) Hybrid systems: computation and control. HSCC 2008. Lecture notes in computer science, vol. 4981. Berlin: Springer, 2008, pp.215–228.
9.
GoffauxGWouwerABernardO.Improving continuous-discrete interval observers with application to microalgae-based bioprocesses. J Process Contr2009; 19: 1182–1190.
10.
StoicanFOlaruSBitsorisG.Controlled invariance-based fault detection for multisensory control systems. IET Control Theory A2013; 7: 606–611.
11.
HagemannW.Efficient geometric operations on convex polyhedra, with an application to reachability analysis of hybrid systems. Math Comput Sci2015; 9: 283–325.
12.
XueBEaswaranAChoN, et al. Reach-avoid verification for nonlinear systems based on boundary analysis. IEEE T Automat Contr2016; 62: 3518–3523.
13.
ZhengGEfimovDPerruquettiW.Design of interval observer for a class of uncertain unobservable nonlinear systems. Automatica2016; 63: 167–174.
14.
PicoHNVAliprantisDC. Voltage ride-through capability verification of DFIG-based wind turbines using reachability analysis. IEEE T Energy Conver2016; 51: 1387–1398.
15.
MeslemN.New idea to design linear interval observers. In: Proceedings of the 3rd IEEE international conference on control, decision and information technologies (CoDIT’16), Malta, 6–8 April 2016, pp.1–6. New York: IEEE.
16.
CombastelC. A state bounding observer based on zonotopes. In: Proceedings of European control conference, Cambridge, 1–4 September 2003, pp.2589–2594. New York: IEEE.
17.
PolyakBTNazinSADurieuC, et al. Ellipsoidal parameter or state estimation under model uncertainty. Automatica2004; 40: 1171–1179.
18.
LeVTHStoicaCAlamoT, et al. Zonotopic guaranteed state estimation for uncertain systems. Automatica2013; 49(1): 3418–3424.
19.
BenChabane SStoicaManiu CAlamoT, et al. Improved set-membership estimation approach based on zonotopes and ellipsoids. In: Proceedings of the IEEE European control conference, Strasbourg, 24–27 June 2014, pp.993–998. New York: IEEE.
20.
CombastelC.Zonotopes and Kalman observers: gain optimality under distinct uncertainty paradigms and robust convergence. Automatica2015; 55: 265–273.
21.
LoukkasNMartinezJMeslemN. Set-membership observer design based on ellipsoidal invariant sets. In: IFAC 2017 world congress, Toulouse, 9–14 July 2017. Geneva: IFAC.
22.
MeslemNLoukkasNMartinezJJ.Using set invariance to design robust interval observers for discrete-time linear systems. Int J Robust Nonlin2018; 28(11): 3623–3639.
23.
PuigVQuevedoJEscobetT.Robust fault detection approaches using interval models. In: Proceedings on IFAC world congress, Barcelona, 21–26 July 2002. Geneva: IFAC.
24.
PuigVStancuAEscobetT, et al. Passive robust fault detection using interval observers: application to the DAMADICS benchmark problem. Control Eng Pract2006; 14(6): 621–633.
25.
IngimundarsonABravoJMPuigV, et al. Robust fault detection using zonotope-based set-membership consistency test. Int J Adapt Control2009; 23: 311–330.
26.
XuFPuigVOcampo-MartinezC, et al. Actuator-fault detection and isolation based on set-theoretic approaches. J Process Contr2014; 24: 947–956.
27.
RaissiTEfimovDZolghadriA.Interval state estimation for a class of nonlinear systems. IEEE T Automat Contr2012; 57: 260–265.
28.
MazencFDinhTNNiculescuSI.Interval observers for discrete-time systems authors. Int J Robust Nonlin2014; 24: 2867–2890.
29.
CacaceFGermaniAManesC.A new approach to design interval observers for linear systems. IEEE T Automat Contr2015; 60: 1665–1670.
30.
DegueKHEfimovDNyJL, et al. Interval observers for secure estimation in cyber-physical systems. In: Proceedings of the IEEE conference on decision and control (CDC), Miami Beach, FL, 17–19 December 2018, pp.4559–4564. New York: IEEE.
31.
RabehiDMeslemNRamdaniN. Secure interval observer for linear continuous-time systems with discrete measurements subject to cyber-attacks. In: Proceedings of the 4th conference on control and fault tolerant systems (SysTol), Casablanca, Morocco, September 2019, pp.336–341. New York: IEEE.
32.
BemporadAGarulliA.Output-feedback predictive control of constrained linear systems via set-membership state estimation. Int J Control2000; 73: 655–665.
33.
LeVTHAlamoCSTCamachoEF, et al. Zonotopes: from guaranteed state-estimation to control. London: Wiley-ISTE, 2013.
34.
QiuQYangFZhuY, et al. Output feedback model predictive control based on set-membership state estimation. IET Control Theory A2020; 14(4): 558–567.
35.
KurzhanskiAVaraiyaP.Ellipsoidal techniques for reachability analysis: internal approximation. Syst Control Lett2000; 41(3): 201–211.
GirardA. Reachability of uncertain linear systems using zonotopes. In: MorariMThieleL (eds) Hybrid systems: computation and control. HSCC 2005. Lecture notes in computer science, vol. 3414. Berlin: Springer, pp.291–305.
38.
NedialkovNSJacksonKR.An interval Hermite–Obreschkoff method for computing rigorous bounds on the solution of an initial value problem for an ordinary differential equation. Reliab Comput1999; 5: 289–310.
39.
RamdaniNMeslemNCandauY.A hybrid bounding method for computing an over-approximation for the reachable space of uncertain nonlinear systems. IEEE T Automat Contr2009; 54(10): 2352–2364.
40.
HuJYangYLiuH, et al. Non-fragile set-membership estimation for sensor-saturated memristive neural networks via weighted try-once-discard protocol. IET Control Theory A2020; 14: 1671–1680.
41.
MartinezJJLoukkasNMeslemN.H-infinity set-membership observer design for discrete-time LPV systems. Int J Control2020; 93(10): 2314–2325.
42.
MüllerM.Uber das fundamentaltheorem in der theorie der gewöhnlichen differentialgleichungen. Math Z1926; 26: 619–645.
43.
WalterEKiefferM. Interval analysis for guaranteed nonlinear parameter estimation. In: Proceedings 13th IFAC symposium on system identification, Rotterdam, 27–29 August 2003, pp.259–270. Geneva: IFAC.
44.
KiefferMWalterESimeonovI. Guaranteed nonlinear parameter estimation for continuous-time dynamical models. In: Proceedings 14th IFAC symposium on system identification, Newcastle, NSW, Australia, 29–31 March 2006, pp.843–848. Geneva: IFAC.
SmithH.Monotone dynamical systems: an introduction to the theory of competitive and cooperative systems (Mathematical surveys and monographs), vol. 41. Providence, RI: American Mathematical Society, 1995.
MeslemNRamdaniN.Interval observer design based on nonlinear hybridization and practical stability analysis. Int J Adapt Control2011; 25(3): 228–248.
53.
MeslemNRamdaniNCandauY.Guaranteed parameter set estimation with monotone dynamical systems using hybrid automata. Reliab Comput2010; 14(10): 88–104.
54.
AstolfiDJungersMZaccarianL.Output injection filtering redesign in high-gain observers. In: Proceedings of the 16th European control conference, ECC’18, Limassol, Cyprus, 12–15 June 2018. New York: IEEE.
55.
AstolfiDMarconiL.A high-gain nonlinear observer with limited gain power. IEEE T Automat Contr2015; 60: 3059–3064.
KiefferMWalterE. Guaranteed nonlinear state estimation for continuous-time dynamical models from discrete-time measurements. In: Proceedings 6th IFAC symposium on robust control, Toulouse, 5–7 July 2006, pp.685–690. Geneva: IFAC.
58.
AngeliDSontagED. Multi-stability in monotone input/output systems. Syst Control Lett2004; 51: 185–202.
59.
ThronC.. The secant condition for instability in biochemical feedback control—parts I. The role of cooperativity and saturability. B Math Biol1991; 53: 383–401.
60.
SontagE.Molecular systems biology and control. Eur J Control2005; 11(4–5): 396–435.
TysonJOthmerH.The dynamics of feedback control circuits in biochemical pathways. In: RosenRSnellFS (eds) Progress in theoretical biology, vol. 5. New York: Academic Press, 1978, pp.1–62.
63.
DochainD.State and parameter estimation in chemical and biochemical processes: a tutorial. J Process Contr2003; 13: 801–818.
64.
JacaquezJSimonC.Qualitative theory of compartmental systems. SIAM Rev1993; 35: 43–79.
65.
AngeliDSontagED. Oscillations in I/O monotone systems. IEEE T Circuits Syst Spec Issue Syst Biol2008; 55: 66–176.
66.
ChambonEBurlionLApkarianP.Overview of linear time-invariant interval observer design: towards a non-smooth optimisation-based approach. IET Control Theory A2016; 10: 1258–1268.
