Stabilizing a class of nonlinear parameter-varying systems using interval observer based on a piecewise framework. (English) Zbl 1531.93343
Summary: This article addresses an interval observer-based control for stabilizing a class of nonlinear parameter-varying systems with noisy output by designing a switching surface. An input-dependent interval observer is firstly developed to estimate the lower and upper bounds of the states. Next, a switching-based controller is designed to stabilize the interval observer which implies the stability of the main parameter-varying system. The developed stabilizing switching surfaces are designed based on the outputs of the main system and the bounds of the states of the observer. By choosing an appropriate piecewise Lyapunov function, the closed-loop stability analysis of the interval observer system leads to a set of linear matrix inequalities including stability and Metzler constraints, simultaneously. The effectiveness of the proposed method is verified using the simulation results.
{© 2022 John Wiley & Sons Ltd.}
{© 2022 John Wiley & Sons Ltd.}
MSC:
| 93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |
| 93C10 | Nonlinear systems in control theory |
| 93B53 | Observers |
Keywords:
interval observer; linear matrix inequality; observer-based control; parameter-varying systems; stabilizing switching surfaceReferences:
| [1] | LuenbergerDG. Observing the state of a linear system. Proceedings of the IEEE Transactions on Military Electronics, Vol. 8, 1964:74‐80; IEEE. |
| [2] | WelchG, BishopG. An introduction to the Kalman filter. Proceedings of the Siggraph Course. Los Angeles, CA, USA; 2001. |
| [3] | CastilloA, GarcíaP, SanzR, AlbertosP. Enhanced extended state observer‐based control for systems with mismatched uncertainties and disturbances. ISA Trans. 2018;73:1‐10. |
| [4] | GuN, WangD, PengZ, LiuL. Observer‐based finite‐time control for distributed path maneuvering of underactuated unmanned surface vehicles with collision avoidance and connectivity preservation. IEEE Trans Syst Man Cybern Syst. 2021;51(8):5105‐5115. |
| [5] | AlwiH, EdwardsC. Robust fault reconstruction for linear parameter varying systems using sliding mode observers. Int J Robust Nonlinear Control. 2014;24(14):1947‐1968. · Zbl 1302.93059 |
| [6] | WeiX, VerhaegenM. LMI solutions to the mixed ℋ−/ℋ∞ fault detection observer design for linear parameter‐varying systems. Int J Adapt Control Signal Process. 2011;25(2):114‐136. · Zbl 1213.93054 |
| [7] | GouzéJL, RapaportA, Hadj‐SadokMZ. Interval observers for uncertain biological systems. Ecol Modell. 2000;133(1-2):45‐56. |
| [8] | MazencF, DinhTN, NiculescuSI. Interval observers for discrete‐time systems. Int J Robust Nonlinear Control. 2014;24(17):2867‐2890. · Zbl 1305.93041 |
| [9] | MeslemN, LoukkasN, MartinezJ. Using set invariance to design robust interval observers for discrete‐time linear systems. Int J Robust Nonlinear Control. 2018;28(11):3623‐3639. · Zbl 1398.93052 |
| [10] | LiJ, WangZ, ZhangW, RaïssiT, ShenY. Interval observer design for continuous‐time linear parameter‐varying systems. Syst Control Lett. 2019;134:104541. · Zbl 1428.93023 |
| [11] | MeslemN, MartinezJ, RamdaniN, BesançonG. An H_∞ interval observer for uncertain continuous‐time linear systems. Int J Robust Nonlinear Control. 2020;30(5):1886‐1902. · Zbl 1465.93079 |
| [12] | RabehiD, MeslemN, RamdaniN. Finite‐gain L_1 event‐triggered interval observers design for continuous‐time linear systems. Int J Robust Nonlinear Control. 2021;31(9):4131‐4153. · Zbl 1526.93160 |
| [13] | RamiMA, ChengCH, dePradaC. Tight robust interval observers: an LP approach. Proceedings of the 47th IEEE Conference on Decision and Control (CDC); 2008:2967‐2972; IEEE. |
| [14] | ThabetREH, RaïssiT, CombastelC, EfimovD, ZolghadriA. An effective method to interval observer design for time‐varying systems. Automatica. 2014;50(10):2677‐2684. · Zbl 1301.93035 |
| [15] | MazencF, BernardO. Asymptotically stable interval observers for planar systems with complex poles. IEEE Trans Autom Control. 2010;55(2):523‐527. · Zbl 1368.93602 |
| [16] | MoisanM, BernardO. Robust interval observers for global Lipschitz uncertain chaotic systems. Syst Control Lett. 2010;59(11):687‐694. · Zbl 1216.34050 |
| [17] | WangX, Pin TanC, WangY, ZhangZ. Active fault tolerant control based on adaptive interval observer for uncertain systems with sensor faults. Int J Robust Nonlinear Control. 2021;31(8):2857‐2881. · Zbl 1526.93038 |
| [18] | RotondoD, CristofaroA, JohansenTA, NejjariF, PuigV. Robust fault and icing diagnosis in unmanned aerial vehicles using LPV interval observers. Int J Robust Nonlinear Control. 2019;29(16):5456‐5480. · Zbl 1430.93150 |
| [19] | LamouchiR, RaïssiT, AmairiM, AounM. Interval observer framework for fault tolerant control of linear parameter‐varying systems. Int J Control. 2018;91(3):524‐533. · Zbl 1396.93041 |
| [20] | WangX, TanCP, LiuL, QiQ. A novel unknown input interval observer for systems not satisfying relative degree condition. Int J Robust Nonlinear Control. 2021;31(7):2762‐2278. · Zbl 1526.93081 |
| [21] | PangB, ZhangQ. Interval observers design for polynomial fuzzy singular systems by utilizing sum‐of‐squares program. IEEE Trans Syst Man Cybern Syst. 2018;50(6):1999‐2006. |
| [22] | ShammaJS, CloutierJR. Gain‐scheduled missile autopilot design using linear parameter varying transformations. J Guid Control Dyn. 1993;16(2):256‐263. |
| [23] | WeiX, delReL. Gain scheduled H∞ control for air path systems of diesel engines using LPV techniques. IEEE Trans Control Syst Technol. 2007;15(3):406‐415. |
| [24] | daSilva CamposVC, NguyenAT, PalharesRM. Adaptive gain‐scheduling control for continuous‐time systems with polytopic uncertainties: An LMI‐based approach. Automatica. 2021;133:109856. · Zbl 1480.93221 |
| [25] | MazencF, BernardO. ISS interval observers for nonlinear systems transformed into triangular systems. Int J Robust Nonlinear Control. 2014;24(7):1241‐1261. · Zbl 1287.93080 |
| [26] | ChenJ, ZhangW, CaoY, ChuH. Observer‐based consensus control against actuator faults for linear parameter‐varying multiagent systems. IEEE Trans Syst Man Cybern Syst. 2017;47(7):1336‐1347. |
| [27] | EfimovD, RaïssiT, ZolghadriA. Stabilization of nonlinear uncertain systems based on interval observers. Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (CDC‐ECC); 2011:8157‐8162; IEEE. |
| [28] | deSouzaAR, EfimovD, RaïssiT. Robust output feedback MPC for LPV systems using interval observers. IEEE Trans Automat Control. 2021. doi:10.1109/TAC.2021.3099449 · Zbl 1537.93241 |
| [29] | EfimovD, RaïssiT, ZolghadriA. Control of nonlinear and LPV systems: interval observer‐based framework. IEEE Trans Automat Control. 2013;58(3):773‐778. · Zbl 1369.93546 |
| [30] | CaiX, LvG, ZhangW. Stabilisation for a class of non‐linear uncertain systems based on interval observers. IET Control Theory Appl. 2012;6(13):2057‐2062. |
| [31] | EfimovD, RaïssiT, PerruquettiW, ZolghadriA. Design of interval observers for estimation and stabilization of discrete‐time LPV systems. IMA J Math Control Inf. 2016;33(4):1051‐1066. · Zbl 1397.93030 |
| [32] | HeZ, XieW. Control of non‐linear systems based on interval observer design. IET Control Theory Appl. 2017;12(4):543‐548. |
| [33] | FaraminM, RezaieB, RahmaniZ. Robust integral feedback control based on interval observer for stabilizing parameter‐varying systems. IET Control Theory Appl. 2019;13(15):2455‐2464. |
| [34] | ChebotarevS, EffimovD, RaïssiT, ZolghadriA. On interval observer design for a class of continuous‐time LPV systems. IFAC Proc Volumes. 2013;46(23):68‐73. |
| [35] | EfimovD, FridmanL, RaïssiT, ZolghadriA, SeydouR. Interval estimation for LPV systems applying high order sliding mode techniques. Automatica. 2012;48(9):2365‐2371. · Zbl 1257.93090 |
| [36] | WilloughbyRA. The inverse M‐matrix problem. Linear Algebra Appl. 1977;18(1):75‐94. · Zbl 0355.15006 |
| [37] | ChangXH, ParkJH, ZhouJ. Robust static output feedback H∞ control design for linear systems with poly‐topic uncertainties. Syst Control Lett. 2015;85:23‐32. · Zbl 1322.93047 |
| [38] | HirschMW, SmithH. Monotone dynamical systems. Handbook of Differential Equations: Ordinary Differential Equations. Vol 2. North‐Holland; 2006:239‐357. · Zbl 1094.34003 |
| [39] | JohanssonM, RantzerA. Computation of piecewise quadratic Lyapunov functions for hybrid systems. IEEE Trans Automat Control. 1998;43(4):555‐559. · Zbl 0905.93039 |
| [40] | HassibiA, BoydS. Quadratic stabilization and control of piecewise‐linear systems. Proceedings of the American Control Conference (ACC); 1998:3659‐3664; IEEE. |
| [41] | WeiY, JingX, ZhengWX, QiuJ, KarimiHR. A new design of asynchronous observer‐based output‐feedback control for piecewise‐affine systems. IEEE Control Syst Lett. 2019;3(2):338‐343. |
| [42] | FengG. Controller design and analysis of uncertain piecewise‐linear systems. IEEE Trans Circuits Syst I Fundam Theory Appl. 2002;49(2):224‐232. · Zbl 1368.93195 |
| [43] | SamadiB, RodriguesL. A unified dissipativity approach for stability analysis of piecewise smooth systems. Automatica. 2011;47(12):2735‐2742. · Zbl 1235.93224 |
| [44] | LofbergJ. YALMIP: A toolbox for modeling and optimization in MATLAB. Proceedings of the IEEE International Symposium on Computer Aided Control Systems Design; 2004:284‐289; IEEE. |
| [45] | SturmJF. Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optim Methods Softw. 1999;11(1-4):625‐653. · Zbl 0973.90526 |
| [46] | BoydSP, El GhaouiL, FeronE, BalakrishnanV. Linear Matrix Inequalities in System and Control Theory. Vol 15. SIAM; 1994. · Zbl 0816.93004 |
| [47] | WangY, XieL, deSouzaCE. Robust control of a class of uncertain nonlinear systems. Syst Control Lett. 1992;19(2):139‐149. · Zbl 0765.93015 |
| [48] | IoannouPA, SunJ. Robust Adaptive Control. Vol 1. PTR Prentice‐Hall; 1996. · Zbl 0839.93002 |
| [49] | DinhQ, GumussoyS, MichielsW, DiehlM. Combining convex-concave decompositions and linearization approaches for solving BMIs, with application to static output feedback. IEEE Trans Automat Control. 2012;57(6):1377‐1390. · Zbl 1369.90170 |
| [50] | El GhaouiL, BalakrishnanV. Synthesis of fixed‐structure controllers via numerical optimization. Proceedings of 1994 33rd IEEE Conference on Decision and Control, Lake Buena Vista, FL, USA, vol. 3; 1994:2678‐2683; IEEE. |
| [51] | HassibiA, HowJ, BoydS. A path‐following method for solving BMI problems in control. Proceedings of the 1999 American Control Conference San Diego, CA, USA, vol. 2; 1999:1385‐1389; IEEE. |
| [52] | ChenJ, LinC, ChenB, WangQG. Output feedback control for singular Markovian jump systems with uncertain transition rates. IET Control Theory Appl. 2016;10(16):2142‐2147. |
| [53] | LiuL, WeiX, LiuX. LPV control for the air path system of diesel engines. Proceedings of the 2007 IEEE International Conference on Control and Automation; 2007:873‐878; IEEE. |
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