Interval Observers Design for Non-Stationary Systems

The paper is devoted to the problem of interval observers design for technical systems described by non-stationary linear dynamic equations under unmatched disturbances and measurement noise. The problem is to design the observer with fewer dimensions than that of the original system; such an observer must generate upper and lower bounds of admissible values of the prescribed linear function of the original system state vector. To construct the interval observer, the reduced-order model of the original system insensitive to the disturbances is designed. It is assumed that the reduced-order model is realized in the diagonal Jordan canonical form. The main advantage of such a form is that it has the main properties which are necessary to the interval observer design and to take into account non-stationarity of the system. As a result, the model is stationary and nonlinear. Besides, such a model allows to reduce limitations on the system under which the interval observer can be designed. The interval observer consists of two subsystems: the first one generates the lower bound, the second one the upper bound. The relations describing both subsystems are given. To take into account the nonlinearities, the notion of monotony of the output variables entering in the nonlinear term on the model is introduced. This notion allows finding out how the upper and lower bounds of these variables will appear in the interval observer. To reduce the interval width, the sliding mode observer is suggested to use. Such an observer is intended to estimate the value of the external disturbances; this estimate is used then in the interval observer to compensate the disturbances. Theoretical results are illustrated by practical example of the electric servoactuator for which the interval observer is designed.

1. Zuev A. V., Zhirabok A. N., Filaretov V. F., Protcenko А. А. Fault identification in non-stationary systems based on sliding mode observers, Mekhatronika, Avtomatizatsiya, Upravlenie, 2021, vol. 22, no. 12, pp. 625—633 (in Russian).

2. Zhirabok A. N., Zuev A. V., Kim C., Bobko E. Yu. Interval observer design for discrete-time nonlinear dynamic systems, Mekhatronika, Avtomatizatsiya, Upravlenie, 2023, vol. 24, no. 6, pp. 283—291 (in Russian).

3. Efimov D., Raissi T. Design of interval state observers for uncertain dynamical systems, Autom. Remote Control, 2015, vol. 77, pp. 191—225 (in Russian).

4. Khan A., Xie W, Zhang L., Liu L. Design and applications of interval observers for uncertain dynamical systems, IET Circuits Devices Syst., 2020, no. 14, pp. 721—740.

5. Kremlev A., Chebotarev S. Synthesis of interval observers for linear systems with variable parameters, Izv. Vuzov. Priborostroenie, 2013, vol. 56, no. 4, pp. 42—46 (in Russian).

6. Efimov D., Raissi T., Perruquetti W., Zolghadri A. Estimation and control of discrete-time LPV systems using interval observers, Proc. 52nd IEEE Conf. On Decision and Control. Florence, Italy, 2013, pp. 5036—5041.

7. Chebotarev S., Efimov D., Raissi T., Zolghadri A. Interval observers for continuous-time LPV systems with L1/L2 performance, Automatica, 2015, vol. 51, pp. 82—89.

8. Liu L., Xie W. Khan A., Zhang L Finite-time functional interval observer for linear systems with uncertainties, IET Control Theory and Applications, 2020, vol. 14, pp. 2868—2878, DOI: 10.1049/iet-cta.2020.0200.

9. Marouani G., Dinh T., Raissi T., Wang X., Messaoud H. Unknown input interval observers for discrete-time linear switched systems, European J. Control, 2021, vol. 59, pp. 165—174, DOI: 10.1016/j.ejcon.2020.09.004.

10. Zhirabok A., Zuev A., Filaretov V., Shumsky A., Kim C. Interval observers for continuous-time systems with parametric uncertainties, Autom. Remote Control, 2023, vol. 84, no. 11, pp. 1137—1147.

11. Misawa E., Hedrick J. Nonlinear observers — a state of the art. Survey, J. Dynamic Systems, Measurements and Control, 1989, vol. 111, pp. 344—352.

12. Zhang K., Jiang B., Yan X., Edwards C. Interval sliding mode observer based fault accommodation for non-minimum phase LPV systems with online control allocation, Int. J. Control, 2019, DOI: 10.1080/00207179.2019.1687932.

13. Zhirabok A., Zuev A., Filaretov V., Shumsky A., Kim C. Jordan canonical form in the diagnosis and estimation problems, Autom. Remote Control, 2022, vol. 83, no. 9, pp. 1355—1370.

14. Wang X., Tan C., Zhou D. A novel sliding mode observer for state and fault estimation in systems not satisfying matching and minimum phase conditions, Automatica, 2017, vol. 79, pp. 290—295.